Pierrick Tranouez
email: pierrick.tranouez@univ-rouen.fr

Éric Daudé
email: eric.daude@univ-rouen.fr

Patrice Langlois
email: patrice.langlois@univ-rouen.fr

† Pierrick

A MULTIAGENT URBAN TRAFFIC SIMULATION

Keywords: Multiagent, urban traffic model, simulation, crisis management, risk management, emergence, complex systems

We built a multiagent simulation of urban traffic to model both ordinary traffic and emergency or crisis mode traffic.

This simulation first builds a modeled road network based on detailed geographical information. On this network, the simulation creates two populations of agents: the Transporters and the Mobiles. Transporters embody the roads themselves; they are utilitarian and meant to handle the low level realism of the simulation. Mobile agents embody the vehicles that circulate on the network. They have one or several destinations they try to reach using initially their beliefs of the structure of the network (length of the edges, speed limits, number of lanes etc.). Nonetheless, when confronted to a dynamic, emergent prone environment (other vehicles, unexpectedly closed ways or lanes, traffic jams etc.), the rather reactive agent will activate more cognitive modules to adapt its beliefs, desires and intentions. It may change its destination(s), change the tactics used to reach the destination (favoring less used roads, following other agents, using general headings), etc. We describe our current validation of our model and the next planned improvements, both in validation and in functionalities.

Introduction

One of the major interrogations in a catastrophe is: will people react correctly to preserve their own security, and doing so, preserve the security of all? This question is most of the time linked with the idea of risk culture and risk training, which are supposed to produce safe reactions at the right time. Is risk education therefore a key factor to preserve life? Thinking so, we assume that the description of behaviors in risk situations should be engaged in term of personal dispositions, which are mostly related to education. In society, norms such as the highway code, or safety instructions in risk situations, are supposed to govern citizen's behaviors. Do circumstances or local contexts produce different behaviors than those acquired, and then alter global vulnerability? The relation between inherited-based behaviors, which can be viewed here as a "follow the rule" behavior, and the circumstances-based behavior, is an interesting perspective to study the question of the vulnerability of populations.

Space is an important factor of risks situations, not only as a support of activities and populations, but also as an actor in itself of the situation. Risk is space related. One defines risk as a probability of space-time interactions between a source (ex: an industrial plant) and a target (ex: an inhabitant building) [START_REF] Daudé | Spatial risks and complex systems: methodological perspectives[END_REF][START_REF] Daudé | A multiagent urban traffic simulation. Part II: dealing with the extraordinary[END_REF]. Space can produce many different contexts throughout its organization. The main difficulty to characterize and manage the risk is then the huge amount of interactions that links entities. Risk has spatial impacts : consider the real estate value near or far from a chemical factory. Risk management transforms the environment, it can be perceived (e.g. type of allowed constructions, fences etc.).

Risk sometimes outlast their management : contaminated soil can remain contaminated decades after the factory generating the contamination has disappeared, along with its most active risk management. This is all the more important in urban area which concentrate a high number of activities, and then reveals some tensions between them. The sharing of a same resource, space, implies some regulations tools in term of laws, of infrastructures and of behaviors. Risks are multi-layered and imply different kinds of actors, human and non-human. A catastrophe in an urban context implies a large number of people and groups, each endowed with their own skills, behaviors and resources. Efforts of public services made in order to mitigate the outcome of the catastrophe are different from, sometime even in contradiction with, those followed by individuals to save their life, but they occur at the same time and the same place. And the same person can respond differently to an event, depending of the context. Furthermore, risks are dynamic, feedbacks and nonlinearities are important. One can observe a domino effect as an explosion in one site produces secondary accidents in the neighborhood, due to the high concentration of activities.

The MOSAIIC project [START_REF] Tranouez | A multiagent urban traffic simulation. Part I: dealing with the ordinary[END_REF][START_REF] Daudé | A multiagent urban traffic simulation. Part II: dealing with the extraordinary[END_REF] aims to observe and understand local and global effects of individual behaviors in the dynamic of a transportation network system after an industrial accident. Few researches take into account the behaviors of group or individual when studying the risk at the scale of a city. Physical aspects override the measure of risk and population damage is a result of these major forces. In this way, intensity of a toxic cloud or of earthquake defines buffers that are used to estimate the number of inhabitants and equipment implicated in the event, and then gives an estimation of the vulnerability of populations. Traditionally, the population dynamics, its ability to move during a catastrophic event are not considered to evaluate vulnerability of this population! When human behaviors are investigated in a risk situation, it is mostly at a very fine scale, for example rooms or building [START_REF] Helbing | Simulating dynamical features of escape panic[END_REF], and one kind of behavior is particularly studied: panic and escaping [START_REF] Helbing | Simulation of pedestrian crowds in normal and evacuation situations[END_REF]. When macro scale is preferred, human behaviors are replaced by traffic behavior. Macroscopic models consider parameters such as traffic density or traffic flow to compute road capacity and then the distribution of traffic in the road network. In risk management, many studies have shown that early stages of the phenomenon are critical on the level of the global damage. This situation appears for example when the diffusion of panic between some individuals produce a snowball effect on the entire population. This social contagion assumes hypnotic effects and selfish behaviors [START_REF] Brown | Social psychology[END_REF]. But individual behaviors in risk situations are not limited to panic and escaping behaviors, even not to selfish behavior. If one considered Bhopal (1984) or Toulouse (2001) accidents, the number of victims or the time of resilience of the system have largely increased due to curiosity: in some circumstances, people want to see the damage. In some other circumstances, one can observe altruistic behavior when self-centered interest would have been expected [START_REF] Clarke | The mist of panic[END_REF]. The MOSAIIC project aims at modelling and simulating all this complexity. We first have to construct a model of urban traffic which is able to capture the urban pulsation, i.e. the main circulation flows of the city during specific time periods. This first step in the modeling process is supposed to be the more parsimonious as possible in term of related social data. Our hypothesis is that we can use the structure of the network and the norms (highway code) to deduce the flow between different points in space and time. In this agent-based model [START_REF] Wooldridge | An Introduction to MultiAgent Systems[END_REF], agents-drivers are supposed to behave as reactive agents, following the rules related to their plans. The second step is to produce a cognitive architecture for agent-drivers to manage specific context: congestion, rerouting, actions due to a major accident, in fact everything which is susceptible to modify its initial plan.

Dealing with human behaviors in a traffic model

If traffic modeling is nearing a century of age, most of these models belong to Operational Research (OR) problems -finding an optimal solution balancing various constraints. In these models, roads and road users were abstracted and aggregated, so as to become a flow problem that could then be optimized. They can answer interesting questions in urban or public transport planning [START_REF] May | Traffic Flow Fundamentals[END_REF].

Sometimes considering average response to a problem is not enough for the scientific problem at hand. We are interested in a dynamic modeling of urban traffic. In this kind of problem, the actions of a few can have a definite impact on the global traffic. An accident implicating half a dozen vehicles in a strategic crossroads of a town can create a traffic jam wave that can affect thousands of vehicles. This is the kind of complex phenomenon we would like to be able to model and simulate. Classic OR tools are not well suited to the task.

Although we are not the first to make this statement [9], models that tried to alleviate this too-large-scale limitation, have mainly tried to use cellular automata for the task. They added some level of individual-based components to their modeling, but still failed to encompass all that could be needed. Cellular automata are eulerian methods -intelligence is in one place, rules describe the behavior of bits of space. Values linked to the cells seem to simulate the entities of the modeled system, the same way alternatively lit crystals in an LCD display can give the illusion an object moves around a screen. This contrasts with lagrangian descriptions, where entities of an environment are distinguished, and their spatial coordinates are but one of their describing characteristics. Unlike what can be easily simulated in a CA, lagrangian entities have a trajectory: even in a discretized space a la CA, they can for example act according to something that happened n time steps and m space steps before or away, or according to a plan. This cannot practically be done in a CA. Multiagent systems belong to this latter category of modeling. As we try to build a model with a grain fine to the level of geometrically correct individual vehicle behavior, from which at least town-quarters-level flow disturbance can arise, we believe this technique is the right one for the task. We then have to build a urban network structure to support these mobile agents.

3 From geometry to topology

Geographical databases

A Geographical Information System (GIS) is a system designed for creating, storing, analyzing and managing spatial data and associated attributes. Although it contains a relational database, it needs to go beyond what is needed for classical alphanumerical databases to manage geometrical information, which is continuous by nature, as opposed to the discreteness of usual databases. Indeed for example the database cannot contain all the points of two segments in order to compute a possible intersection: other storing and managing methods must be used for the geometric data of the system.

A geographic database is generally comprised of layers or coverage overlapping on a same spatial domain. Each layer contains homogeneous spatial features such as the limits of a city, the course of a river, the geometry of a road etc. Each feature is described in two different ways. First the geometric and optionally topological information is stored in different binary files in the base. Second the record description is a line in the record table; it contains different attributes and descriptions of the feature (generally text or numbers).

ESRI Shapefiles

The first step of the constitution of our system is the constitution of a basic layer of geographic database. This layer is built from the importation of shapefiles, a GIS file format popularized by ESRI [START_REF]ESRI Shapefile Technical Description[END_REF]. In order to build a traffic simulation, we will build our model from data relative to the road network and optionally from other localized information such as living or working areas. A shapefile is mainly constituted of three files: one contains the attribute table (.dbf), another contains the geometric data (.shp) and the third is an index allowing matching entries of the first with those of the second.

A shapefile contains only the geometric description of objects through a collection of 2D or 3D coordinates that represents, according to the layer type, a cloud of points, open polygon lines for networks or closed polygon lines to describe the boundary of surfaces. The topological information, which describes in geomatics the relationships between the geometric entities, such as connections of edges with nodes in a graph or the adjacency between zones in a surface partition, is absolutely not present in a shapefile, and must therefore be computed by our application form the raw geometry of the imported data.

To build a realistic representation of the traffic network of an important urban agglomeration able to simulate the circulation of tens of thousand of vehicles, we had to conceive a network layer structure both complex and efficient. Furthermore, as the importation of data was coming from existing data provider such as IGN, NAVTEQ or Tele Atlas, we had to deal with the way each of them modeled things in their solutions.

Urban network structure specifications

A road network is modeled according to specifications that are in part common to any network and in part dependent on decisions made according to the data

General specifications

A road network shares the properties of any geographic network. It is constituted of two main geometric entities: lines, linear components, comprised of several shape points, and nodes, point components that join or terminate lines.

These two entities are joined in an oriented multigraph G = (S, A, f ) where S is the set of vertices, associated to the geometric nodes, while the set A of edges is associated to the geometric lines, while function f : A → S × S associates to each edge one initial and one final vertex.

Unlike most other geographic information layer, a road network may not be planar: two lines can intersect in their planar projection without modeling an intersection in the real world. This happens when these lines are at different altitudes such as in bridges, tunnels, or motorway embranchments.

Furthermore, geographic graph are topological graph, which differ from usual graphs in that they are associated to one geometric representation, called the embedding of the graph. Only vertices of degree 3 or more are considered to be true vertex, those of degree 2 being seen as shape points, useful for the geometric information they bring, but not "true" connectors. The geometric representation of the graph is always present to the mind of the geographer, which may create misunderstandings with other scientists more used to a more abstract representation of graph, with planar graph rather than plane graph. As previously said, it is also sometimes extended to non planar graph: the geometric information in the shapefile represents in that case the projection on a connected compact 2-manifold of a graph embedded in a connected compact 3-manifold [START_REF] Preparata | Computational Geometry: An introduction[END_REF] (intuitively: a 3D graph is drawn on a surface).

The attribute table associated to the network will contain all the traffic related information, such as the number of lanes, speed limits, sense of travel etc. Nonetheless this information may not be associated to elementary lines or nodes. For example major roads may contain different lines and important roundabout may contain different nodes and lines. We therefore defined the notion of super-nodes that relate to several nodes (and the assorted sub-graph) and super-edges that relate to several edges (and the assorted sub-graph). G is therefore a hypergraph in these conditions. Whether these conditions are met or not depend on modeling decisions made by the data provider.

Geographic data based specifications

There are different ways of structuring the geographic information in a shapefile to model a network. For example NAVTEQ chose in its Navstreets product to create a node for each intersecting link, even if the road they model are not connected. Another layer represents the relative elevation of the entities of this first layer. Both must therefore be used to correctly build the road network in our simulation. Another example is the orientation of the edges, as the links are oriented following another convention (called Reference nodes) than what could be used in a shapefile, and the edge must therefore be computed following this convention.

Building the topology from the geometry

Building a topology from the geometric information contained for example in a shapefile depends on the kind of spatial organization we want to represent.

Planar mesh

In the case of a surface mesh (ex: limits of countries, of urban areas, of town quarters etc.), we aim at rebuilding the boundaries and the junction nodes between them from closed polygonal chains (a.k.a. polylines). The layer we produce is thereafter structured around a planar multigraph of vertices, edges and faces, and with each oriented edge associated to 2 vertices (initial and terminal) and to 2 faces (left and right).

The building algorithm uses a quadtree and a tree connecting each point, in which all the points of the shapefile are organized. Each leaf of the quadtree contains a point P i and 4 branches for the 4 quadrants of space (NE, SE, SW, NW) surrounding P i . This structure allows for a quick detection of the multiplicity of points. For example, a point with a multiplicity of 3 or more will be associated to a vertex, while a point of a multiplicity of 2 will be a shape point of an edge. Furthermore, the connection tree allows the quick detection of adjacent points along a polyline, and detecting the superposition of two lines forming the boundaries of two zones, or the succession of angular sectors around a vertex common to three polygons or more.

Networks

In order to build the structure of a planar network (for example hydrographic or of roads), we do not store faces but the polar order of succession in the edges. Each edge stores the next edge turning left and the prior edge turning right. This structure is known as DCEL, Doubly Connected Edge List [START_REF] Preparata | Computational Geometry: An introduction[END_REF]. The algorithm to generate this topology uses the dynamic quadtree structure used that was used to build the DCEL. The road network often exists in 3D, although despite the existence of this possibility in the format specification, most shapefiles only contain a 2D geometric representation. The data provider must in that case model the altitude differently, and our algorithm must be adapted to this. For example NAVTEQ's Navstreets [START_REF][END_REF] uses another layer called z-levels that must be consulted to know whether a point corresponds to a node or not.

At the end of this step, we have a topological graph that is structured like the road network, but without its semantics. We will now build from it and from the database part of the shapefile a non-topological graph that models this ontology.

3.5 From static topology to trafficoriented

Traffic oriented graph

Our traffic model is individual-based: each vehicle will be modeled as an agent. This implies the creation of an adapted environment for them, in terms both suitable to their ontology and adapted to the geographic data we reaped. For that a graph will be built, a transport graph that will contain the necessary structures and values. This first version of our models is only interested in simulating motor vehicle: pedestrians and bicycle are ignored.

The database contains the sense of travel and the traffic restrictions for each topological edge. One oriented edge is created for each sense of direction allowed for motor vehicles. Edges and vertices of the transport graph are called elements. To each element is associated a data container and a vehicle transporter.

The data associated to an edge are for example its geometric length, its number of lanes, its speed limits etc. The data associated to a vertex is notably the size of the container of its transporter, depending on the number and the sizes of the edges connected to him.

Transporters are non-mobile agents associated to elements. They handle parts of the collective behavior of the vehicles. They will be described in more depth in the following section.

Routes in the graph

Mobile agents will try to reach destinations in the graph.

As we intend to simulate a realistic traffic of tens of thousand of vehicles, we want to facilitate their computing of their trajectory. To do that, we build a set of "shortest" path stored in the traffic graph. We compute a weight on the edges that combines different parts of its data: its length, the speed it can reasonably be driven upon, its estimated width based on the number of lanes etc. to model the attractiveness of this edge. After that we compute Dijkstra's algorithms [START_REF] Dijkstra | A note on two problems in connexion with graphs[END_REF] from each vertex to all the others, which we store in each vertex. This data takes numberOf V ertices 2 bytes of data, which is important, but allows the computation of a good path by an agent in constant time, which is a good thing as hundreds of agents are generated at all time in the simulation (simulating vehicles entering the road network of the simulated urban agglomeration).

Mobile agents on the network

Our agents are mainly so far car agents, trying to go from one place to another.

Strategic behavior

Modeling in details the various detailed trajectories of car users is a research problem in itself [START_REF] Banos | Thévenin Création de champs de potentiel et simulation d'itinéraires à partir de l'enquête ménages-déplacement In: Information Géographique et Dynamique Urbaine[END_REF]. Nonetheless we are not interested in who did what or why, but only in what are the fluxes in our network in typical scenarios. When an agent is injected in the network, a starting point and a destination are randomly chosen. This randomness is not necessarily uniform. If we suppose the agglomeration centered on its main town, like the agglomeration of Rouen that we simulated more than others, we can shape different distribution, favoring the likelihood of drawing rather a inner or an outer edge for example. Traffic between 8:00 AM and 9:00 AM for example starts mainly on the border or outside the agglomeration and ends to the same distance to the center (outer edges): we can simulate traffic that do that. When shops close in the town center, we have a traffic that is mainly outer bound, with a more important center generation: we can simulate that. We do not have to know what this car and its driver did in the morning, we don't have to simulate realistically its history, as long as we model the actual traffic fluxes right. Once the agent knows where it is, and what its destination is, it can use the best paths stored in the traffic network to plan a trajectory. It then drives here, adapting his path through its tactical behavior, and managing its immediate surroundings through its operational behavior.

Operational behavior

The planned trajectory of an agent is a succession of edges. Once in an edge the agent tries to drive to its end, the next connection, where it will be able to choose the next planned edge. When it enters an edge, the agent first chooses a lane if several are available, based on the traffic density in each, with a bias for the rightmost lane. As we have a good geometric description of the lane, the driving behavior is fairly detailed, incorporating the length of the car, its capacity/willingness to accelerate and brake, the taste of its driver for long/short safety distance, its taste for following or breaking speed limits etc. All this is incorporated in a driving model inspired by Martin Treiber's Intelligent Driver Model [START_REF] Kesting | General lane-changing model MOBIL for car-following models[END_REF]. IDM is a longitudinal traffic model, so we had to expand it to handle multiple lanes and crossroads (the original IDM works for an unlimited one-way, one-lane road) ; we did not use Treiber's MOBIL lane changing model as it is better adapted to motorways than to urban lane changing decisions. The data provided by geographic providers does not include right of passage or traffic lights at crossroads. We therefore had to develop our own model aiming at the simulation of crossroads in a heavy traffic. When a vehicle reaches a crossroad, it slows down and acts according to the fluidity of traffic in the crossroad, in the edge it is currently upon and in the edge it whishes to go to. If they are encumbered, it will more often wait in its way, but it may enter the crossroad and wait here, thus encumbering it (with a more or less strong individual tendency to do so). If the edge it is aiming at has multiple lanes, it will watch both of them, to see if it could fit in one.

Tactical behavior

Although vehicles have an original plan, they will adapt it to what they perceive of their environment. When stuck in what they perceive is a jam, they will try to find alternate routes out of it to their destination.

The first method we used is the simpler one. When a vehicle doesn't move enough to its liking -this saturation is variable amongst agents -it tries to take alternate paths as soon as possible, favoring the roads with least dense circulation -although this is not absolute, so as to avoid loops. Once it estimates it's far enough from the jam that sprang this alternate behavior, it resumes its standard behavior, using the best path table to find a suitable one to its destination. The second one is more sophisticated, as it will have uses beyond mere traffic avoidance. Its intelligence is modeled more in the Transporter agents than in the vehicles.

Transporters estimate their encumbrance. To do that, they employ direct measure -how many vehicles do they contain over how many vehicles can they contain in average -but also statistics on the proportion of vehicles they contain that are annoyed by the traffic -as described in the first method -and information from the Transporters around them. If based on this they decide they are encumbered they also warn the Transporters around them of their perception. This will lower the threshold for them to feel encumbered. Once encumbered, the nodes they are connected to will recompute their best path table, using a huge weight for the encumbered edges. When a vehicle arrives to one of these nodes and wants to go to one of the jammed edges, it is informed of the edge state, and it can recompute a route around it, or take the edge anyway. This mechanism is also theoretically interesting, as it is an implementation of an emergent property: the interactions of individual behavior affect the behavior of an agent of an higher scale, who alters his behavior, which in turns transforms the behavior of the lower level vehicles. This reifies the perception an individual driver can have of the state and dynamics of the traffic he is plunged in as a whole.

A Transporter can also be barred, because of an accident for example. In that case the same mechanism is used, except that this time circumnavigating is mandatory. The mechanism of these two states is especially useful in what was the original purpose of our model and its main application: simulating urban important accidents -such as industrial accident -as the modeler can bar the edges it wants as part of his scenario, and see how the traffic adapts to it in simulation real time, as the vehicles discover the evolving road network and fluxes.

Unexpected individual behaviors in risk situations

MOSAIIC is about situations where contextual mobility can occur and can diffuse or have large consequences on the global circulation. We call contextual mobility a mobility which is associated to short-range goals (for example to avoid a crowd or a jam) and whose results differ from the initial planning (for example changing the planned trajectory). We will now consider an urban industrial accident. This accident has a finite extension area and a well-determined intensity, represented by a buffer. Inside this buffer, a proportion of people, related to intensity for example, want to change their initial plan. Outside this buffer, behaviors are less reactive. Some want to escape, others want to see and for others "show must go on", and they want to follow their way. In this example we see that the same person might have different behaviors depending on its location related to the event, or related to the behaviors of its neighbors.

Contextual mobility

We have then defined different kinds of behaviors and methods related to different short-term goals:

• Chicken behavior: the goal is to find the opposite direction of the source (the buffer), and to drive following this way;

• Spectator behavior: the goal is to find the source of danger and to go there. If agent is already in the place, then he stays here;

• Pragmatic behavior: here the agent selects a new destination in the network and tries to reach it. This behavior simulates the fact that some people will want to reach their children at school or husband or wife at their working place;

• Wandering behavior: there is no goal, this behavior is the sign of distress. At each time step, just select a road and go there.

• Roadrunner behavior: this method consists in always selecting the less congested road and to go there. This method can be connected to the Chicken or Bystander behavior;

• Sheep behavior: here agent follows the crowd whatever the direction.

We will discuss now how the behaviors themselves can be implemented, not why or when one or the other will be chosen.

Behaviors classification

In order to implement them, we will distinguish three categories of behavior: global, planar and local. These categories are based on the actual behavior, and not on the motivations behind it.

A global behavior is one that makes a reasoning about the road network. Pragmatic behavior falls in this category: the agent will try to find a good path to his newly decided destination using his knowledge of the network. Spectator can also fall here.

A planar decision also chooses a destination but tries to reach it using orientation as if no roads existed, as if the vehicle was on an open plan. Of course the network will offer constraints, but a general cardinal like direction will guide the agent. Chicken and possibly Spectator will fall in this category. This means there are two possible implementations of this behavior.

A local decision is one based on local-only data: Wandering, Roadrunner and Sheep will fall there.

Class implementation in the simulation

Global behaviors are implemented in the agents to allow them to reach their initial destination. The motivations change between those two cases, but the underlying mechanics stay the same.

Local require little complexity. Wandering is trivial, Roadrunner and Sheep differ only by the sign of their optimization. We also implemented a simple anti-loop measure: Roadrunners for example will choose the less congested road unless they already went recently through this one.

Planar require the ability to choose an edge out of a node based on a global direction. Depending on what the modeler desires, he can choose a distance from the current road intersection, and the agent will choose the intersection at less than the selected distance (expressed in Euclidean distance or number of edges in a path leading to it) that is the closest to the desired direction. An antiloop measure can be added.

The behaviors previously described are ways of coping with an extraordinary situation. Most urban important accidents will have their consequences felt locally at the beginning, before it spreads. The evolution of the perturbation will be like waves spreading from the initial locus outward. If the extraordinary behaviors are the waves, the metaphorical medium of this propagation is the ordinary traffic flow. We therefore need a sophisticated modeling of the day-to-day activities of vehicles in an urban agglomeration as we have seen before.

Example

Figure 2 depicts an example of the distribution of vehicles in the main roads of the city. Starting at this point, we generate an event in the city that is supposed to represent an accident. This event, for the purpose of the simulation, is perceived by all individuals and is considered as a repulsive event. In figure 3, this event is a mouse-click event located by the user without any consideration about the reality of the area. As agents perceived the impact zone (in fact its euclidean coordinates), they all change their planned trajectory. Once in a crossroad, all mobiles pick out the Chicken behaviors and pick an edge, choosing the one whose other node forms the widest angle (as close to π as possible) with their current node and the coordinates of the explosion. This escaping behavior is for instance not applied in concurrence with any other mobility strategies or tactical behavior: they have not the possibility to avoid traffic jam or loops. The main effect of the general application of this rule is purely the draining of the transportation network. Of course this "Hollywood panic" scenario is not relevant in reality but let us test implemented mechanisms. Vulnerability increases when a certain quantity of actors changed their dynamics of mobility, mainly after a shift in their goals. Beliefs, desires and goals are then important to take into account in this kind of model.

Discussion

One of the problems for the validation is that modeling as seldom been taken to such a detail level. This level is A PhD student has recently started a study to increase our knowledge of the traffic in Rouen agglomeration, and thus give us more data to compare our simulation to.

Further developments

We have defined methods modeling mobility itself, but we now need to model the decision processes for picking or switching between these methods. In an ordinary situation, people follow their own planning and most of the times never deviate of their schedule. But how to justify and explain the fact that in some circumstances, people shift from one behavior to another, from an ordinary behavior to one of the extraordinary described here such as Sheep or Roadrunner? We have so far described models that have been implemented, simulated and whose results are being investigated. We will now describe what we are currently building and improving : providing some cognitive abilities to our agents to model their alternative strategical and tactical decision making. We use here a Beliefs -Desires -Intentions (BDI) [START_REF] Wooldridge | Reasoning About Rational Agents[END_REF] description of the decision process of our agents. Our BDI implementation is simple and homemade : although we need to model some complexity of the decision making, what we need to get right is the behaviour of a large crowd of agents, not necessarily be psychologically realistic at the agent level.

• Beliefs here model inner and outer perception (am I happy or annoyed, is the circulation around me fluid or jammed), and also acts as a short term memory :

Crisis source : If I am in a crisis mode, what caused it, where was it and when did I learn about it ?

Anti-loop : have I used this road recently ?

Behavior satisfaction : Has my annoyance grown or shrunk since I started using this behavior ?

• Desires are a list of destinations. Our mobile agents always want to go somewhere. Initially they have a list of destination (dropping the children at school, fetching the laundry, going to work). This list can be altered in case of a crisis ; ex : I stop trying to go to work, go to my spouse job place to get her then head to my children school, and finally flee.

• Intentions mode the plans of action of our agents.

They know where they want to go, they have a perception of their mental states and their surroundings, now how do they reach their goals ? The tools we give them for that is a possibility of switching their tactical behaviors (cf. 4.3), changing their destinations, and change their saturation (should I often try new tactic or stick to one for a longer time?).

One of the main questions is then: how fine a knowledge and understanding of people mental procedures is it necessary to have to simulate crowd dynamics and vulnerability of transportation network? In other words, what level of detail is needed in the modeling of individual agent to accurately model the behavior of a crowd of them?

Figure 1 :

 1 Figure 1: 3 different scenarios of source/destination pseudo-random choice

Figure 2 :

 2 Figure 2: An example of traffic in a town before an accident occurs.

Figure 3 :Figure 4 :

 34 Figure 3: The same traffic after the accident occurs (red circle). The agents are all adopting Chicken behavior (in blue).

Figure 5 :

 5 Figure 5: Beliefs, Desires and Intentions of our agents.

Acknowledgements

The authors would like to thank the GRR SER and the region Haute-Normandie for the funding of the MOSAIIC program from which this work stems from.

The authors would like to thank the GRR SER and the region Haute-Normandie for the funding of the MOSAIIC program from which this work stems from.


F Barthe

B Huou

Perimeters, uniform enlargement and high dimensions

We study the isoperimetric problem in product spaces equipped with the uniform distance. Our main result is a characterization of isoperimetric inequalities which, when satisfied on a space, are still valid for the product spaces, up a to a constant which does not depend on the number of factors. Such dimension free bounds have applications to the study of influences of variables.

Introduction

Let (X, d, µ) denote a metric probability space, where X is separable and µ is a Borel probability measure on (X, d). For a Borel subset A of X, we define, for r > 0, the open r-neighbourhood of A by A r = x ∈ X d(x, A) < r , and its outer and inner boundary measures (also called Minkowski contents) by

µ + (A) = lim inf r→0 + µ (A r ) -µ (A) r , µ -(A) = µ + (X \ A).
The isoperimetric problem consists in obtaining sharp lower bounds on the above quantities in terms of the measure µ(A). The isoperimetric function of (X, d, µ), denoted by I (X,d,µ) (or simply I µ when there is no ambiguity on the underlying metric space), is defined for p ∈ [0, 1] as follows :

I µ (p) = inf A⊆X; µ(A)=p min µ + (A), µ -(A) (1)

= inf

A⊆X; µ(A)∈{p,1-p}

µ + (A) (2)
where the infimum is taken over all Borel subsets A of X. As we can see from the definition, I µ is the largest function such that, for every A ⊆ X, µ + (A) ≥ I µ (µ(A)) and for every t ∈ [0, 1], I µ (t) = I µ (1t). Notice also that I µ (0) = I µ (1) = 0.

Given metric probability spaces (X i , d i , µ i ), i = 1, . . . , n, several metric structures can be considered on the product probability space

(X 1 × • • • × X n , µ 1 ⊗ • • • ⊗ µ n ).
Throughout this paper, we equip this product with the supremum distance d = d

(n)

∞ defined by d (n) ∞ (x 1 , . . . , x n ), (y 1 , . . . , y n ) := max i d i (x i , y i ).

We shall also say that d

(n)

∞ is the ℓ ∞ -combination of the distances d i , 1 ≤ i ≤ n. The isoperimetric problem has been intensively studied in the Riemannian setting, where the geodesic distance on a product manifold is the ℓ 2 -combination of the geodesic distance on the factors. Hence, from a geometric viewpoint, the choice of the ℓ ∞ -combination is less natural than the one of the ℓ 2 -combination d

(n) 2 ((x i ) n i=1 , (y i ) n i=1 ) = i d(x i , y i ) 2 1/2 .
Nevertheless, the study of the uniform enlargement has various motivations. We briefly explain some of them.

Firstly the isoperimetric problem for the uniform enlargement is technically easier to deal with in the setting of product spaces, due to the product structure of metric balls. This often allows to work by comparisons. For instance Bollobás and Leader study this problem for the uniform measure on the cube in order to solve the discrete isoperimetric problem on the grid [START_REF] Bollobás | Edge-isoperimetric inequalities in the grid[END_REF]. Since d

(n) ∞ ≤ d (n) 2 ≤ √ n d (n)
∞ , it easily follows that

1 √ n I (X n ,d (n) ∞ ,µ n ) ≤ I (X n ,d (n) 2 ,µ n ) ≤ I (X n ,d (n) ∞ ,µ n ) .
This approach was used e.g. by Morgan [START_REF] Morgan | Isoperimetric estimates in products[END_REF] for products of two Riemannian manifolds.

Another motivation for studying the isoperimetric problem for the uniform enlargement is that it amounts to the study of the usual isoperimetric problem for a special class of sets. Let us explain this briefly in the setting of R n equipped with a probability measure dµ(x) = ρ(x)dx and the ℓ ∞ distance. If ρ is continuous and A ⊂ R n is a domain with Lipschitz boundary, its outer Minkowski content is

µ + (A) = ∂A n A (x) 1 ρ(x)dH n-1 (x),
where n A (x) is a unit outer normal to A at x (unit for the Euclidean length). Consequently, the boundary measure for the uniform enlargement coincides with the usual one ∂A ρ(x)dH n-1 (x), for sets A such that almost surely on ∂A the outer normal is equal to a vector of the canonical basis of R n (or its opposite). These so-called rectilinear sets comprise cartesian products of intervals I 1 × • • • × I n , their finite unions and their complements. Hence the isoperimetric problem for the uniform enlargement is closely connected to the usual isoperimetric problem restricted to the class of rectilinear sets (actually, a smooth domain A can be approximated by rectilinear sets in such a way that their boundary measures approach the one of A for the uniform enlargement). Note that rectilinear sets naturally appear when studying the supremum of random variables, as {x ∈ R n max i x i ∈ [a, b]} is rectilinear. This was one of the original motivations of Bobkov and Bobkov-Houdré [START_REF] Bobkov | Isoperimetric problem for uniform enlargement[END_REF][START_REF] Bobkov | Weak dimension-free concentration of measure[END_REF] for studying isoperimetry for the uniform enlargement.

Eventually, let us mention that isoperimetric inequalities for the uniform enlargement naturally appear in the recent extension by Keller, Mossel and Sen [START_REF] Keller | Geometric influences[END_REF] of the theory of influences of variables to the continuous setting.

Computing exactly the isoperimetric profile is a hard task, even in simple product spaces (see e.g. the survey article [START_REF] Ros | The isoperimetric problem[END_REF]). However, various probabilistic questions involve sequences of independent random variables and require lower estimates on the isoperimetric profile of n-fold product spaces, which actually do not depend on the value of n. First observe that for all integers n ≥ 1,

I (X n+1 , d (n+1) ∞ , µ n+1 ) ≤ I (X n , d (n) ∞ , µ n ) ,
which holds because for every set

A ⊂ X n , µ n+1 (A × X) = µ n (A) and (µ n+1 ) + (A × X) = (µ n ) + (A).
Therefore one may define the so-called infinite dimensional isoperimetric profile of (X, d, µ) as follows :

for t ∈ [0, 1], I µ ∞ (t) := inf n≥1 I (X n , d (n) ∞ , µ n ) ≤ I (X,d,µ
) . This quantity has been investigated by Bobkov [START_REF] Bobkov | Isoperimetric problem for uniform enlargement[END_REF], Bobkov and Houdré [START_REF] Bobkov | Weak dimension-free concentration of measure[END_REF] and Barthe [START_REF] Barthe | Infinite dimensional isoperimetric inequalities in product spaces with the supremum distance[END_REF]. In particular, Bobkov has put forward a sufficient condition for the equality I µ ∞ = I µ to hold. This condition depends only on the function I µ but it is rather restrictive. However it allowed to get a natural family of isoperimetric inequalities for which there exists K > 1 such that I µ ≥ I µ ∞ ≥ 1 K I µ . We shall say in this case that the isoperimetric inequality with profile I µ tensorizes, up to a factor K.

The goal of this article is to provide a workable necessary and sufficient condition for the latter property to hold. We were inspired by a sufficient condition for tensorization, given by E. Milman [START_REF] Milman | On the role of convexity in functional and isoperimetric inequalities[END_REF] in the setting of ℓ 2 -distances on products. We now describe the plan of the paper. In the next section, we recall the known sufficient condition for I µ ∞ = I µ and propose a new one. Building on this, we provide a sufficient condition for tensorization up to a factor in the third section. By a careful study of product sets, we actually show that this condition is also necessary. The final section draws consequences of our isoperimetric inequalities to the theory of influences of variables : following the argument of [START_REF] Keller | Geometric influences[END_REF], we obtain an extension of the Kahn-Kalai-Linial theorem about the existence of a coordinate with a large influence.

Let us conclude this introduction with some useful notation. If (Y, ρ) is a metric space we define the modulus of gradient of a locally Lipschitz function f : Y → R by :

|∇f |(x) = lim sup ρ(x,y)→0 + |f (x) -f (y)| ρ(x, y) ,
this quantity being zero at isolated points. Note that when the distance is given by a norm on a vector space, that is ρ(x, y) = xy , and when f is differentiable, then the modulus of gradient coincides with Df (x) * . We shall work under the following Hypothesis (H): for every m, n ∈ N * and for every locally Lipschitz function f :

X m+n → R, for µ m+n -almost every point (x, y) ∈ X m × X n : |∇f |(x, y) = |∇ x f |(x, y) + |∇ y f |(x, y).
This assumption holds in various cases : when (X, d) is an open metric subset of a Minkowski space (R n , . ) and when µ is absolutely continuous with respect to Lebesgue's measure, or for Riemannian manifolds when the measure is absolutely continuous with respect to the volume form (as a consequence of Rademacher's theorem of almost everywhere differentiability of Lipschitz functions). On the contrary, this hypothesis often fails in discrete settings.

Sharp isoperimetric inequalities

We start by recalling a couple of important results about extremal half-spaces for the isoperimetric problem. The first one below is due to Bobkov and Houdré [START_REF] Bobkov | Some connections between isoperimetric and Sobolev-type inequalities[END_REF] and deals with the real line. Before stating it, we need to introduce some notations. Let M be the set of Borel probability measures on R which are concentrated on a possibly unbounded interval (a, b) and have a density f which is positive and continuous on (a, b). For µ ∈ M, the distribution function F µ (x) := µ((-∞, x]) is one-to-one from (a, b) to (0, 1) and one may define

J µ (t) = f F -1 µ (t) , t ∈ (0, 1).
We may as well consider J µ as a function on [0, 1] by setting J µ (0) = J µ (1) = 0. The value of J µ (t) represents the boundary measure of the half-line of measure t starting at -∞. Let L ⊂ M denote the set of (non-Dirac) log-concave probability measures on R (the density f is of the form e -c for some convex function c).

Proposition 1 ([6]

). The map µ → J µ is one-to-one between the set M and the set of positive continuous functions on (0, 1). It is also one-to-one between the subset L of log-concave probability measures and the set of positive concave functions on (0, 1). Moreover for µ ∈ M, the following properties are equivalent :

(i) I µ = J µ (meaning for any p ∈ (0, 1), the infimum in (1) is attained for the set -∞, F -1 µ (p) , (ii) the measure µ is symmetric around its median, i.e. J µ is symmetric around 1 2 , and for all p, q > 0 such that p + q < 1,

J µ (p + q) ≤ J µ (p) + J µ (q).
The next basic lemma allows to compare the various conditions on isoperimetric profiles that appear in the rest of the article. In particular, it shows that the above result encompasses a classical theorem of Borell, asserting that for even log-concave probability measures on R, half-lines are solutions to the isoperimetric problem.

Lemma 1. Let T ∈ (0, +∞] and K : [0, T ) → R + be a non-negative function. Consider the following properties that K may verify :

(i) K is concave, (ii) t → K(t)/t is non-increasing, (iii) for all x, y ∈ [0, T ) with a + b < T , it holds K(a + b) ≤ K(a) + K(b).
Then (i) =⇒ (ii) and (ii) =⇒ (iii).

Proof. If K is concave then t → (K(t) -K(0))/t is non-increasing. Since t → K(0)/t is nonincreasing as well, the first implication follows. Assuming (ii) and without loss of generality a ≤ b,

K(a + b) ≤ (a + b) K(b) b = a K(b) b + K(b) ≤ K(a) + K(b).
The next result provides sharp isoperimetric inequalities in high dimensions. It goes back to the dissertation thesis of S. Bobkov. See also [START_REF] Bobkov | Isoperimetric problem for uniform enlargement[END_REF].

Theorem 1. Let J : [0, 1] → R + be a concave function, with J(t) = J(1t) for all t ∈ [0, 1]. Assume that for all a, b ∈ [0, 1], J(ab) ≤ aJ(b) + bJ(a).

(

) 3
Then for every space (X, d, µ) verifying Hypothesis (H),

I µ ≥ J =⇒ I µ ∞ ≥ J.
Moreover there exists an even log-concave probability measure ν on R such that I ν = I ν ∞ = J and for every n, coordinate half-spaces are solutions of the isoperimetric problem for ν n .

Condition (3) may be verified in a few instances as J(t) = t(1t). However, it is not so easy to deal with, in particular in conjunction with the symmetry assumption. For these reasons, stronger conditions of more local nature are useful. In [START_REF] Barthe | Infinite dimensional isoperimetric inequalities in product spaces with the supremum distance[END_REF], it is shown that (3) is verified when J is concave, twice differentiable and -1/J" is concave. Observe that condition (3) amounts to the subadditivity of the function u → e u J(e -u ) on R + . Hence, using the second part of Lemma 1, we obtain that the condition "t → J(t)/(t log(1/t)) is non-decreasing" implies (3) as well. By a tedious but straightforward calculation, this yields a neat variant of one of the main results of [START_REF] Barthe | Infinite dimensional isoperimetric inequalities in product spaces with the supremum distance[END_REF]:

Corollary 1. For β ∈ [0, 1], the function K β defined for t ∈ [0, 1] by K β (t) := t(1 -t) log β 3 t(1 -t) ,
satisfies that for every space (X, d, µ) verifying Hypothesis (H) and all c ≥ 0,

I µ ≥ cK β =⇒ I µ ∞ ≥ cK β .
Let us point out that (3) is not the best sufficient condition for the conclusion of the above theorem to hold. The optimal condition given by Bobkov's approach is the following : for every Borel probability measure N on [0, 1],

J t dN (t) ≤ J(t) dN (t) + 1 0 J N ([0, t]) dt.
Actually when µ ∈ F is a probability measure on R and J = J µ = I µ , it is not hard to check, considering subgraphs, that the above condition is necessary and sufficient for having

I µ = I µ ∞ .
However this condition is hard to verify in practice, and most of the work in Bobov's proof consists in showing that when J is concave, it boils down to (3).

Next, we develop a different approach to dimension free isoperimetric inequalities. We use classical methods to make a link between isoperimetric inequalities, and some Beckner-type functional inequalities, which nicely tensorize.

Lemma 2. Let a ∈ (0, 1] and (X, d, µ) be a metric probability space. Let c > 0, then the following assertions are equivalent :

(i) For all p ∈ [0, 1], cI µ (p) ≥ p -p 1 a , (ii) For every locally Lipschitz function f : X → [0, 1], c |∇f | dµ ≥ f dµ - f a dµ 1 a .
Proof. Assuming (i), we apply the co-area inequality to an arbitrary locally Lipschitz function f (see e.g. [START_REF] Bobkov | Some connections between isoperimetric and Sobolev-type inequalities[END_REF]); next we take advantage of the isoperimetric inequality for µ:

c |∇f | dµ ≥ c 1 0 µ + ({f ≥ t}) dt ≥ 1 0 µ({f ≥ t}) -µ({f ≥ t}) 1 a dt = f dµ - 1 0 µ({f ≥ t}) 1 a dt.
In order to conclude that the second assertion is valid, we apply the Minkowski inequality with exponent 1/a ≥ 1:

1 0 µ({f ≥ t}) 1 a dt a = 1 0 1 f (s)≥t dµ(s) 1 a dt a ≤ 1 0 (1 f (s)≥t ) 1 a dt a dµ(s) = f a dµ.
The fact that the second assertion implies the first one is rather standard : one applies the functional inequalities to Lipschitz approximations of the characteristic function of an arbitrary Borel set A ⊂ X (see Lemma 3.7 in [START_REF] Bobkov | Some connections between isoperimetric and Sobolev-type inequalities[END_REF]). This yields cµ + (A) ≥ µ(A)µ(A) 1 a . Applying the inequality to 1f instead of f and using |∇f | = |∇(1f )| and then taking approximations of

1 A gives cµ + (A) ≥ 1 -µ(A) -(1 -µ(A)) 1 a for all A, which is equivalent to cµ -(A) ≥ µ(A) -µ(A) 1 a
for all Borel sets A.

The following extension of the classical subadditivity property of the variance is due to Latała and Oleszkiewicz [START_REF] Latała | Between Sobolev and Poincaré[END_REF]. It allowed them to devise functional inequalities with the tensorization property. Actually, they focused on Sobolev inequalities involving L 2 -norms of gradients, with applications to concentration inequalities. Here we aim at functional inequalities involving L 1 -norms of gradients and provide information about isoperimetric inequalities. Lemma 3. Let (Ω 1 , µ 1 ) and (Ω 2 , µ 2 ) be probability spaces and let (Ω, µ) = (Ω 1 × Ω 2 , µ 1 ⊗ µ 2 ) be their product probability space. For any non-negative random variable Z defined on (Ω, µ) and having finite first moment and for any strictly convex function φ on [0, +∞) such that 1 φ ′′ is a concave function, the following inequality holds true :

E µ φ(Z) -φ (E µ Z) ≤ E µ E µ 1 φ(Z) -φ (E µ 1 Z) + E µ 2 φ(Z) -φ (E µ 2 Z) .
Theorem 2. Let (X, d, µ) be a metric probability space verifying hypothesis (H). Let a ∈ [ 1 2 , 1] and c > 0. If for all p ∈ (0, 1), I µ ≥ c(pp 1 a ), then for all p ∈ (0, 1),

I µ ∞ (p) ≥ c(p -p 1 a ).
Proof. By Lemma 2, we know that for every locally Lipschitz function f :

X → [0, 1] 1 c |∇f | dµ ≥ f dµ - f a dµ 1 a . ( 4
)
We shall prove that this functional inequality tensorizes, meaning that for all n the same property is verified by µ n . Applying Lemma 2 again will give the claimed dimension-free isoperimetric inequality.

Checking the tensorization property is done along the same lines as in [START_REF] Latała | Between Sobolev and Poincaré[END_REF]. Assume that (X 1 , ν 1 , d 1 ) and (X 2 , ν 2 , d 2 ) satisfy (4). Since a ∈ [ 1 2 , 1], Lemma 3 applies to Φ(t) = t 1 a and gives

f dν 1 dν 2 - f a dν 1 dν 2 1 a = Φ(f a ) dν 1 dν 2 -Φ f a dν 1 dν 2 ≤ Φ(f a ) dν 1 -Φ f a dν 1 dν 2 + Φ(f a ) dν 2 -Φ f a dν 2 dν 1 ≤ 1 c |∇ 1 f | + |∇ 2 f | dν 1 dν 2 ,
where |∇ i f | is the norm of the gradient of f taken with respect to the i-th variable. When ν 1 = µ m and ν 2 = µ n , we may apply Hypothesis (H) to replace the function in the latter integral by the norm of the full gradient |∇f |. This allows to show by induction that for all n, µ n verifies the claimed functional inequality.

This result readily generalizes :

Theorem 3. Let c : [ 1 2 , 1] → R + , and consider for p ∈ [0, 1], L(p) := sup a∈[ 1 2 ,1] c(a) max p -p 1 a , 1 -p -(1 -p) 1 a .
If (X, d, µ) satisfies (H) and I µ ≥ L then

I µ ∞ ≥ L.
Moreover there exists an even probability measure ν on R such that I ν = I ν ∞ = L and such that for all n, coordinate half-spaces are solutions to the isoperimetric problem for ν n .

Proof. Observe that since, by definition, isoperimetric functions of probability measures are symmetric with respect to 1 2 , the property for all p ∈ [0, 1],

I µ ≥ c(p -p 1 a ) is equivalent to I µ (p) ≥ cM a (p), for all p, where M a (p) := max p -p 1 a , 1 -p -(1 -p) 1 a .
Hence the fact that

I µ ≥ L implies I µ ∞ ≥ L is a direct consequence of the previous theorem, applied for all values of a. Next, it is not hard to check that for a ∈ [ 1 2 , 1]
, M a is subadditive, being a supremum of two concave functions defined on [0, 1]. And, since the property "J(x + y) ≤ J(x) + J(y) for all x, y" is stable under supremum, it follows that L is also subadditive.

Hence, by Proposition 1, there exists an even probability measure ν on R such that I ν = L and half-lines solve the isoperimetric problem for ν. As we just proved,

I ν ≥ L ensures that I ν ∞ ≥ L. Combining this with L = I ν ≥ I µ ⊗∞ yields I ν ∞ = L. The coordinate halfspace {x ∈ R n x 1 ≤ t}
has same measure and boundary measure (for ν n ), as the set (-∞, t] (for ν). It is then clear that it solves the isoperimetric problem.

Remark that for a ∈ ( 1 2 , 1), the function

M a (p) = max p -p 1 a , 1 -p -(1 -p) 1 a
is not concave, hence the measure ν a is not log-concave. Actually, M a does not even have its maximum at 1 2 . Hence it cannot be obtained as a supremum of concave functions which are in addition symmetric around 1/2. Therefore it gives a genuinely new example of a measure for which coordinate half-spaces solve the isoperimetric problem in any dimension (that could not be deduced from Theorem 1).

Approximate inequalities

Let us start with some notations. Given two non-negative functions f, g defined on a set S ⊂ R and D ≥ 1, we write f ≈ D g and say that f and g are equivalent up to a factor D if there exists a > 0 such that for all x ∈ S, a g(x) ≤ f (x) ≤ Da g(x). We write f ≈ g when there exists D such that f ≈ D g. We say that a non-negative function f defined on a set S ⊂ R is essentially non-decreasing (with constant D ≥ 1) when there exists a non-decreasing function g on S such that f ≈ D g. In the same way, we may define the notion of essentially non-increasing functions. Also, a non-negative function f defined on an interval is said to be essentially concave (or pseudoconcave) if it is equivalent to a concave function.

The next proposition provides workable formulations of the above definitions. The part about essentially concave functions is due to Peetre [START_REF] Peetre | On interpolation functions[END_REF].

Lemma 4. Let f be a non-negative function defined on S ⊂ R. Then f is essentially non-decreasing (resp. essentially non-increasing) with constant D ≥ 1 if and only if for every s ≤ t in S,

f (s) ≤ Df (t) resp. f (t) ≥ Df (s) .
When f is defined on (0, +∞), the following assertions are equivalent :

(i) f is essentially concave with some constant C 1 , (ii) There exists C 2 ≥ 1 such that for all s, t ∈ R * + , f (s) ≤ C 2 max 1, s t f (

t).

(iii) There exists C 3 ≥ 1 such that on R * + , f is essentially non-decreasing and t → f (t) t is essentially non-increasing, both with constant C 3 .

Moreover, the smallest possible constants verify

C 1 /2 ≤ C 2 = C 3 ≤ C 1 .
Proof. The argument for essentially non-decreasing functions is very simple and we skip it. Let us just point out that it involves the least non-decreasing function above f , which is given by

f (t) := sup{f (x) | x ∈ S ∩ (-∞, t]}.
Next let us focus on concavity issues. The equivalence of the last two statements is obvious. Assume f is essentially concave on R * + . Then there exists a concave function h on R * + which is equivalent to f . And as f is positive, h is positive, therefore, being concave, h is necessarily non-decreasing on R * + . Moreover t → h(t) t is non-increasing on R * + . So f satisfies the third condition. Eventually, let us assume the second condition and show that f is equivalent to a concave function. The natural guess is the least concave majorant of f , which is explicitly given for t > 0 by

f (t) := sup n i=1 λ i f (t i ) n ∈ N * , λ i ≥ 0, t i > 0, n i=1 λ i = 1 and n i=1 λ i t i = t . By definition f ≤ f . Let n ∈ N * , t ∈ R * + , (λ i ) 1≤i≤n and (t i ) 1≤i≤n such that, for all i, λ i ≥ 0, n i=1 λ i = 1 and n i=1 λ i t i = t.
Using the hypothesis, we obtain

n i=1 λ i f (t i ) ≤ C 2 n i=1 λ i max 1, t i t f (t) ≤ C 2 n i=1 λ i + n i=1 λ i t i t f (t) = 2C 2 f (t)
Therefore f ≤ f ≤ 2C 2 f and we have shown that f is essentially concave.

We are now ready to state our main results :

Theorem 4. Let J be a non-negative function defined on [0, 1] with J(0) = 0. Assume that it is symmetric around 1 2 (i.e. for every t ∈ [0, 1] , J(t) = J(1t)) and that the function

t ∈ (0, 1) → J(t) t log(1/t)
is essentially non-decreasing with constant D. Then for every metric probability space (X, d, µ) satisfying Hypothesis (H):

I µ ≥ J =⇒ I µ ∞ ≥ 1 c D J, with c D = 2(D/ log 2) 2 ≤ 5D 2 .
Moreover, there exists a symmetric log-concave measure ν on the real line such that, on [0, 1],

J ≈ I ν ≈ I ν ⊗∞ . If in addition J is concave one can take c D = 2D for D > 1 and c 1 = 1.
Remark 1. This result should be compared to a theorem of E. Milman in [START_REF] Milman | On the role of convexity in functional and isoperimetric inequalities[END_REF], where a similar condition is given for dimension-free isoperimetric inequalities for the ℓ 2 combination of distances on products (in other words for the Euclidean enlargement). His condition involves an essential monotonicity property of J/I γ where γ is the one-dimensional standard Gaussian measure. On (0, 1/2] it is known that I γ (t) ≈ t log(1/t).

In order to formulate a converse statement, we introduce the following hypothesis : we say that (X, d, µ) enjoys the regularity property (R) if for all t ∈ (0, 1), I µ (t) < +∞ and for all n ∈ N * , t ∈ (0, 1) and ε > 0 there exists a Borel set A ⊂ X n with µ n (A) = t, (µ n ) + (A) ≤ I µ n (t) + ε and

(µ n ) + (A) = lim h→0 + µ n (A h \ A) h ,
where the products X n are equipped with the uniform distance. This hypothesis means that there are almost solutions of the isoperimetric problems for which the lim inf in the definition of the Minkowski content is actually a real limit. Thanks to Theorem 15 in [START_REF] Barthe | Log-concave and spherical models in isoperimetry[END_REF] it is not hard to check this property for log-concave measures on the real line. We will give more comments on this hypothesis in Remark 3 below.

Theorem 5. Let (X, d, µ) satisfy hypothesis (R). Then the map

t ∈ (0, 1) → I µ ∞ (t) t log(1/t)
is continuous and essentially non-decreasing.

Combining these two theorems, we can formulate our results as an equivalence : Corollary 2. Let (X, d, µ) denote a metric space equipped with a Borel probability measure µ and satisfying hypothesis (R) and (H). Then the following assertions are equivalent :

(i) There exists a constant C such that Iµ J 1 is essentially non-decreasing on (0, 1) with constant C, (ii) There exists a constant K ≥ 1 such that, on [0, 1],

1 K I µ ≤ I µ ∞ ≤ I µ .
We introduce two functions, both defined on [0, 1] by : J 0 (t) = t and J 1 (t) = t log 1 t . The next lemma gives a different formulation of the main condition appearing in the previous theorems.

Lemma 5. Let K : [0, 1] → R + be a non-negative function such that K is symmetric with respect to 1 2 (i.e. for t ∈ [0, 1], K(t) = K(1 -t)).
Then the following assertions are equivalent :

(i) There is a constant C such that K J 1 is essentially non-decreasing on (0, 1) with constant C.

(ii) There exists constants C 0 and C 1 such that K J 0 is essentially non-increasing on 0, 1 2 with constant C 0 and K J 1 is essentially non-decreasing on 0, 1 2 with constant C 1 .

Moreover, the smallest possible constants verify

C ≤ C 0 C 1 log 2 and C 0 ≤ C log 2 , C 1 ≤ C.
Proof. We use the concavity of the map t → (1t) log 1 1-t , which yields, for every t ∈ 0, 1 2 , t log 2 ≤ (1t) log 1 1-t ≤ t. Assuming (i), K J 1 is essentially non-decreasing on 0, 1 2 with constant C. For the second part of the assertion, let 0 < s ≤ t ≤ 1 2 . Then,

K(t) t ≤ K(1 -t) (1 -t) log 1 1-t ≤ C K(1 -s) (1 -s) log 1 1-s ≤ C log 2 K(s) s . (5)
For the converse implication: assuming (ii), we first check that K J 1 is essentially non-decreasing on 1 2 , 1 . Let

1 2 ≤ s ≤ t < 1, then K(s) s log 1 s ≤ 1 log 2 K(1 -s) 1 -s ≤ C 0 log 2 K(1 -t) 1 -t ≤ C 0 log 2 K(t) t log 1 t .
To get the property on the whole interval (0, 1), it suffices to use 1 2 as an intermediate point. The next corollary describes the possible size of an infinite dimensional isoperimetric profile: Corollary 3. Let (X, d, µ) denote a metric space equipped with a Borel probability measure µ and satisfying hypotheses (R) and (H).

If inf t∈(0, 1 2 ]

I µ (t) t = 0 then I µ ∞ is identically 0, else there exist α, β > 0 such that for all t ∈ [0, 1], α min (t, 1 -t) ≤ I µ ∞ (t) ≤ β min t log 1 t , (1 -t) log 1 1 -t .
Remark 2. The function defined on [0, 1] by t → min (t, 1t) is the isoperimetric function of the double-sided exponential measure on R, e -|x| dx/2. Using the notation and results of Corollary 1, we observe that it is equivalent to the function K 0 (t) = t(1t). Moreover there is a log-concave probability measure ℓ 0 on the real line for which K 0 = I ℓ 0 = I ℓ ∞ 0 (actually, ℓ 0 is the standard logistic measure ℓ with density e -x (1+e -x ) 2 with respect to Lebesgue's measure). Hence the lower bound is optimal up to the multiplicative factor.

The upper bound of I µ ∞ given in the above corollary is due to Bobkov and Houdré [START_REF] Bobkov | Some connections between isoperimetric and Sobolev-type inequalities[END_REF]. A similar remark applies to it: the quantity in the upper estimate is equivalent to the function K 1 of Corollary 1, which is also an infinite dimensional isoperimetric profile (of a measure which is reminiscent of Gumble laws, as its distribution function is of the order of e -βe -y when y → -∞, for some β > 0).

The fact that the infinite dimensional isoperimetric profile is either trivial, or at least as big as the one of the exponential measure was already discovered, in slightly different forms, by Talagrand [START_REF] Talagrand | A new isoperimetric inequality and the concentration of measure phenomenon[END_REF] and by Bobkov and Houdré [START_REF] Bobkov | Weak dimension-free concentration of measure[END_REF].

Proof of Corollary 3. By Theorem 5, there exists C ≥ 1 such that for all t ∈ (0, 1/2],

I µ ∞ (t) ≤ Ct log 1 t × 2 log 2 I µ ∞ 1 2 . ( 6
)
Applying Theorem 5 again, together with Lemma 5, we get that there exists D ≥ 1 such that for all t ∈ (0, 1/2],

D I µ ∞ (t) t ≥ 2I µ ∞ 1 2 .
Therefore, assuming inf t∈(0, Iµ(t) t = 0, and using that I µ ≥ I µ ∞ , we can deduce that I µ ∞ 1 2 = 0. Then [START_REF] Bobkov | Some connections between isoperimetric and Sobolev-type inequalities[END_REF] and the symmetry of isoperimetric functions yield I µ ∞ = 0 pointwize.

Next assume that there exists κ > 0 such that I µ (t) ≥ κt for all t ∈ (0, 1/2]. Then Theorem 4 applies to J(t) := κ min(t, 1t) (Lemma 5 gives a quick way to check the hypothesis) and gives I µ ∞ ≥ cJ for some c > 0.

Remark 3. Our results are stated for general metric spaces, but are devised for continuous settings (e.g. for which the values taken by the measure cover all [0, 1]). This is why additional hypotheses appear in our statements. One may find Hypothesis (H) quite natural (it is related to a.e. differentiability of Lipschitz functions). On the other hand, Hypothesis (R) is more demanding, as it seems to require approximation theorems by smooth sets.

Let us point out a possible variant of Theorem 5 where all the hypotheses are incorporated in the structure of the ambient space: assume that X is a finite dimensional vector space of dimension p, that the distance d is induced by a norm N on X and that µ has a positive C 1 density h with respect to Lebesgue's measure, µ = h.L p . We equip the product spaces X n with d ∞ , the ℓ ∞ -combination of N , i.e. for x, y ∈ X n , d ∞ (x, y) = max 1≤i≤n N (x i , y i ). Then, instead of using the Minkowski content as a definition of the boundary measure, let us chose the notion of generalized perimeter instead : if A ⊆ X n is measurable, then

P µ ⊗n ,∞ = sup A n i=1 |∇ i (ϕh)| dL np ϕ ∈ C 1 c (X n ) and sup x∈X n d ∞ (ϕ(x), 0) ≤ 1 ,
where |∇f | is the modulus of gradient of f . Since the perimeter is defined as a supremum (recall that the Minkowski content is an inferior limit), the proof of Lemma 8 below does not require any regularity assumption. Hence the proof of Theorem 5 applies without any changes and does not require (R). The proof of Theorem 4 also applies to this new setting, without assuming (H), with the following main modification: instead of using functional inequalities for locally Lipschitz functions, we work in the class of functions of bounded variations. We refer the reader to the book of Ambrosio, Fusco and Pallara [START_REF] Ambrosio | Functions of bounded variation and free discontinuity problems[END_REF] for an exhaustive study of this approach in the Euclidean case. This requires to use various results about these functions: co-area inequality (Theorem 3.40), approximation by smooth functions (Theorem 3.9), approximate differentiability (Theorem 3.83 and Proposition 3.92 among others.

Proof of Theorem 4

We start with a few preliminary statements.

Lemma 6. Consider a function K : [0, 1] → R + with K(0) = 0. Assume that K is symmetric with respect to 1 2 and that K J 1 is essentially non-decreasing on (0, 1) with constant D. Then (i) K is essentially non-decreasing on [0, 1 2 ] with constant 2D e log 2 , (ii) K is essentially concave. More precisely there exists a concave function I : [0, 1] → R + , which is symmetric with respect to 1 2 , and is equivalent to K up to a factor 2D/ log 2. Proof. Observe that the function J 1 (t) = t log(1/t) is increasing on (0, 1/e] and decreasing on [1/e, 1). Its maximum is therefore J(1/e) = 1/e.

Assume that 0 ≤ s ≤ t ≤ 1 2 . Then, by hypothesis K(s) ≤ D J 1 (s) J 1 (t) K(t). If t ≤ 1 e , we can conclude that K(s) ≤ DK(t). If t ∈ (1/e, 1/2], we argue differently

K(s) ≤ D J 1 (s) J 1 (t) K(t) ≤ D J 1 (1/e) J 1 (1/2) K(t) = 2D e log 2 K(t).
This concludes the proof of (i).

Next, let us prove (ii). Consider the map K defined on R + by :

K(t) = K(t) if t ∈ 0, 1 2 K 1 2 if t ≥ 1 2
Combining (i) and the second part of (ii) in Lemma 5, one readily checks that K satisfies the hypothesis of Assertion (iii) in Lemma 4 with constant D log 2 (≥ 2D e log 2 ). Hence there exists a concave function H which is equivalent to K on (0, +∞), up to a factor 2D/ log 2. Define I to be the restriction of H to 0, 1 2 , extended at 0 by I(0) = 0 and to [0, 1] by symmetry with respect to 1 2 . Since H is concave and non-negative on (0, +∞), it is also non-decreasing. Therefore, the function I is concave as well. As K ≈ H on (0, +∞), we obtain by restriction that K ≈ I on (0, 1/2], up to the same constant. Since K(0) = I(0) = 0, and both I and K are symmetric with respect to 1/2, we can conclude that I ≈ K on [0, 1], up to a factor 2D/ log 2.

The following result shows how we exploit the essentially monotonicity properties of J/J 0 and J/J 1 where J 0 (t) = t and J 1 (t) = t log(1/t). Proposition 2. Let J : (0, 1) → R + such that for all t, J(t) = J(1t). Assume that on (0, 1/2], J/J 0 is essentially non-increasing with constant D 0 and J/J 1 is essentially non-decreasing with constant D 1 . Then there exists a function c : [1/2, 1) → R + such that for all t ∈ (0, 1),

J(t) ≥ sup a∈[1/2,1) c(a) t -t 1 a ,
and for all t ∈ (0, 1 2 ],

J(t) ≤ 2D 0 max(D 0 , D 1 ) sup a∈[1/2,1) c(a) t -t 1 a .
The proof of this proposition relies on the following statement, which is related to [4, Lemma 19]. Lemma 7. Let Φ : 0, 1 log 2 → R + . If Φ is essentially non-increasing with constant C 0 , then for all y ∈ (0, 1 2 ], sup

α∈(0,1] Φ(α) 1 -y α ≥ 1 2C 0 Φ 1 log(1/y) .
If, in addition, s → sΦ(s) is essentially non-decreasing with constant C 1 , then for all y ∈ (0, 1/2],

sup

α∈(0,1] Φ(α) 1 -y α ≤ max(C 0 , C 1 ) Φ 1 log(1/y) .
Proof. In order to bound the supremum from below, we just select an appropriate value for α: if y ∈ (0, 1/e], choosing α = 1/ log(1/y) ∈ (0, 1], we get (1e -c )Φ c log(1/y) .

sup α∈(0,1] Φ(α) 1 -y α ≥ (1 -e -1 )Φ 1 log(1/y) . If y ∈ [1/e,
For c ≥ 1, we know that

Φ c log(1/y) ≤ C 0 Φ 1 log(1/y)
and we bound 1e -c from above by 1. For c ∈ (0, 1], we take advantage of the hypothesis on x → xΦ(x), in the form cΦ(cx) ≤ C 1 Φ(x) for x > 0:

(1 -e -c )Φ c log(1/y) ≤ C 1 1 -e -c c Φ 1 log(1/y) ≤ C 1 Φ 1 log(1/y) .
These two estimates readily give the claim.

Proof of Proposition 2. For α ∈ (0, 1/ log 2], we define

Φ(α) := J e -1/α e -1/α = J J 0 e -1/α .
Since e -1/α ∈ (0, 1/2], our hypothesis ensures that Φ is essentially non-increasing with constant D 0 . Notice that

αΦ(α) := J e -1/α e -1/α log 1 e -1/α = J J 1 e -1/α
Hence by hypothesis, it is essentially non-decreasing with constant D 1 . Therefore we may apply the previous lemma to Φ. Since by definition, Φ(1/ log(1/y)) = J(y)/y, it gives that for all y ∈ (0, 1/2],

J(y) y ≥ 1 max(D 0 , D 1 ) sup α∈(0,1] Φ(α) 1 -y α ,
and for all y ∈ (0, 1/2],

J(y) y ≤ 2D 0 sup α∈(0,1] Φ(α) 1 -y α .
Multiplying these inequalities by y, and setting a := 1/(1 + α), the former estimate gives for y ∈ (0, 1/2]

J(y) ≥ 1 max(D 0 , D 1 ) sup α∈(0,1] Φ(α) y -y 1+α = 1 max(D 0 , D 1 ) sup a∈[1/2,1) Φ 1 a -1 y -y 1 a . ( 7
)
Hence we have prove the claimed lower bound on J with c(a) = Φ(a -1 -1)/ max(D 0 , D 1 ). We proceed in the same way with the upper bound on J. The ratio of the upper bound to the lower bound is 2D 0 max(D 0 , D 1 ). It remains to extend the lower bound [START_REF] Bobkov | Weak dimension-free concentration of measure[END_REF] to values y ∈ (1/2, 1). To do this we use the symmetry of J and the fact that for all a ∈ [1/2, 1] and all s ∈ [1/2, 1), 1s -(1s) 1 a ≥ ss 1 a (this follows from the comparison of second derivatives, observing that equality holds at 1/2 and 1): for y ∈ (1/2, 1),

J(y) = J(1 -y) ≥ c(a) 1 -y -(1 -y) 1 a ≥ c(a) y -y 1 a .
Proof of Theorem 4. Let us denote by D 0 ≥ 1 the smallest constant such that J/J 0 is essentially non-increasing on (0, 1/2] with constant D 0 . Similarly let D 1 ≥ 1 be the smallest constant such that J/J 1 is essentially non-decreasing on (0, 1/2] with constant D 1 .

First, we apply Proposition 2. With the notation of the proposition, it follows that for all a ∈ [1/2, 1) and all t ∈ [0, 1],

I µ (t) ≥ J(t) ≥ c(a) t -t 1 a .
Note that for t = 0 or t = 1 all quantities vanish. Next, Theorem 2 tell us that I µ ∞ (t) ≥ c(a) t-t 1 a . This is true for all a and all t, hence applying the second part of Proposition 2, we deduce that for all t ∈ [0, 1/2],

I µ ∞ (t) ≥ sup a∈[1/2,1) c(a) t -t 1 a ≥ 1 2D 0 max(D 0 , D 1 ) J(t).
Since both I µ ∞ and J are symmetric with respect to 1/2, we can conclude that for all t ∈ [0, 1], it holds I µ ∞ (t) ≥ J(t)/(2D 0 max(D 0 , D 1 )).

In the general case, we know by Lemma 5 that D 0 ≤ D/ log 2 and D 1 ≤ D. Therefore we get that

I µ ∞ ≥ J/c D with c D = 2D 2 /(log 2) 2 .
In the particular case where J is concave, we know that J(t)/t is non-increasing, so that D 0 = 1 and I µ ∞ ≥ J/(2D 1 ) ≥ J/(2D). The paragraph after Theorem 1 explains that when J is concave, symmetric and such that J/J 1 is non-decreasing, the inequality I µ ≥ J implies I µ ∞ ≥ J. Hence in this case the conclusion of Theorem 4 is valid with c 1 = 1.

Proof of Theorem 5

First, we recall a classical property of infinite dimensional profiles, which comes from testing isoperimetric inequalities on product sets. It was put forward by Bobkov in [START_REF] Bobkov | Isoperimetric problem for uniform enlargement[END_REF]. Lemma 8. Let (X, d, µ) be a metric space equipped with a Borel probability measure µ and satisfying the regularity property (R). Then the infinite-dimensional isoperimetric profile of (X, d, µ) satisfies, for every a, b ∈ [0, 1] :

I µ ∞ (ab) ≤ aI µ ∞ (b) + bI µ ∞ (a). ( 8
)
Proof. The inequality is obvious if a or b is equal to 1, or to 0 since I µ ∞ (0) = 0. Let a, b ∈ (0, 1) and ε > 0. Let m, n ∈ N * . Let A ⊂ X m be any set with µ m (A) = a. Let B ⊂ X n be any set with

µ n (B) = b, (µ n ) + (B) ≤ I µ n (b) + ε and (µ n ) + (B) = lim h→0 µ n (B h \ B) h ,
which is possible thanks to Hypothesis (R).

Then consider A × B ⊆ R m+n . Obviously, µ m+n (A × B) = ab. The uniform enlargement of a product set is still a product:

(A × B) h = A h × B h for any h > 0. Therefore 1 h µ m+n ((A × B) h \ (A × B)) = µ m (A h )µ n (B h ) -µ(A)µ(B) h = µ m (A h \ A) h µ n (B h ) + µ m (A) µ n (B h \ B) h
Since by hypothesis lim h→0 (µ n (B h )µ n (B))/h < +∞, we know that lim h→0 µ n (B h ) = µ n (B) (note the convergence holds by monotonicity). Taking upper limits in h → 0, and observing that two of the three terms have limits, we deduce from the latter inequality that

(µ m+n ) + (A × B) ≤ (µ m ) + (A) µ n (B) + µ m (A) (µ n ) + (B). (9)
Since I µ m+n (ab) ≤ (µ m+n ) + (A × B), we obtain after optimizing on sets A of measure a and using the hypothesis on the boundary measure of B:

I µ m+n (ab) ≤ I µ m (a) b + a I µ n (b) + ε .
Letting ε tend to 0, and m, n tend to +∞ gives the claim [START_REF] Bollobás | Edge-isoperimetric inequalities in the grid[END_REF].

The symmetry property (I µ ∞ (t) = I µ ∞ (1t) for all t ∈ 0, 1]) and the two-points inequality (8) are enough to deduce Theorem 5, as the next statement shows:

Proposition 3. Let I : [0, 1] → [0, +∞] be an application satisfying that for all a, b ∈ [0, 1] I(a) = I(1 -a) and I(ab) ≤ aI(b) + bI(a), (10)
with the convention that +∞ × 0 = 0. If there exists x 0 ∈ [0, 1] such that lim sup x→x 0 I(x) < +∞ then I is continuous and t → I(t)/(t log(1/t)) is essentially non-decreasing on (0, 1).

The condition of local boundedness around some point cannot be removed as shown by the following example:

I(t) = 0 if t ∈ Q and I(t) = +∞ otherwise.
The proof of the proposition uses the next two easy lemmas. Lemma 9. Let S ⊂ (0, 1) be a set with the following stability property:

x ∈ S and y ∈ S =⇒ 1x ∈ S and xy ∈ S .

If S is not empty then it is dense in (0, 1). Moreover, if S has non-empty interior then S = (0, 1).

In other words, if S is neither ∅ nor (0, 1) then S and (0, 1) \ S are dense in (0, 1). This is the case for instance of S = Q ∩ (0, 1).

Proof. Let t ∈ (0, 1) be an element of S, then for all n ∈ N * , x n := 1t n belong to S and the sequence (x n ) tends to 1. Given 0 < a < b < 1, let us show that there is a point of S between a and b. Choose k large enough such that x k > max(b, a/b). Then for all n ≥ 1, (x k ) n ∈ S. Obviously

x k ≥ b and lim n (x k ) n = 0. Let n 0 be maximal with (x k ) n 0 ≥ b. Then b > x n 0 +1 k = x n 0 k x k > b × a b = a. Hence x n 0 +1 k ∈ S ∩ (a, b).
This completes the proof of the density of S. Assume now that (a, b) ⊂ S for some 0 < a < b < 1. Consider an arbitrary x ∈ (0, 1) and let us show that x ∈ S. If x ∈ (a, b), we have nothing to prove. If x ∈ (0, a], we use the fact that S being non-empty is dense: there exists y ∈ S ∩ (x/b, x/a). Since S contains y and (a, b), the stability by product ensures that S also contains (ya, yb). Hence x ∈ (ya, yb) ⊂ S. Eventually, if x ∈ [b, 1), we consider 1x ∈ (0, 1b]. By the symmetry assumption (1b, 1a) ⊂ S, so the latter argument yields 1x ∈ S, and using symmetry again x ∈ S.

The next lemma is a classical result about subadditive functions on R + (see e.g. [START_REF] Kuczma | An introduction to the theory of functional equations and inequalities[END_REF]) : Lemma 10. Let K : R + → R be a subadditive function with lim 0 K = 0. Then

lim h→0 + K(h) h = sup t>0 K(t) t • Proof. Denote S = sup t>0 K(t)
t . Given any u < S, there exists

x 0 ∈ R * + such that K (x 0 ) > ux 0 . For any h ∈ (0, x 0 ), write x 0 = nh + δ with n = ⌊ x 0 h ⌋ and δ ∈ [0, h). By subadditivity of K, ux 0 < K(x 0 ) ≤ nK(h) + K(δ) = (x 0 -δ) K(h) h + K(δ).
Next, we let h tend to 0 + . In this case δ → 0 + and K(δ) → 0, hence (for any u < S)

u ≤ lim inf h→0 + K(h) h •
Therefore S ≤ lim inf h→0 + K(h) h . On the other hand, lim sup h→0 +

K(h)

h ≤ S holds by definition.

Proof of Proposition 3.

There is nothing to prove if I is identically 0, so we assume that I does not vanish everywhere. Observe that the two-points inequality in [START_REF] Kalai | Threshold phenomena and influence: perspectives from mathematics, computer science, and economics[END_REF], applied for a = b = 0, yields I(0) = 0. Consider the subset S 1 of (0, 1) of points x such that the function I is bounded on a neighbourhood of x. Our hypothesis lim sup x→x 0 I(x) < +∞ ensures that S 1 has non-empty interior. Thanks to [START_REF] Kalai | Threshold phenomena and influence: perspectives from mathematics, computer science, and economics[END_REF], one readily checks that S 1 is stable by product and by symmetry with respect to 1/2. Hence Lemma 9 applies to S 1 and shows that S 1 = (0, 1). This means that I is locally bounded at every point of (0, 1). By compactness, we deduce that I is bounded on any segment [a, b] ⊂ (0, 1).

The next step of the proof is an argument of Bobkov and Houdré, that we include for completeness. The two-points inequality implies by induction that for all a ∈ [0, 1] and any integer k ≥ 1, I(a k ) ≤ ka k-1 I(a). Let t ∈ (0, 1/e]. Choosing k = ⌊log(1/t)⌋ ≥ 1 and a = t 1/k in the latter inequality leads to

I(t) ≤ kt 1-1 k I t 1 k = t log(1/t) I t 1/k t 1/k • Using that for x ≥ 1, x/⌊x⌋ ∈ [1, 2]
, we obtain that t 1/k = exp(-log(1/t)/⌊log(1/t)⌋) ∈ [e -2 , e -1 ]. Hence for t ∈ (0, e -1 ], I(t) ≤ Ct log(1/t) where C = e 2 sup{I(s); s ∈ [e -2 , e -1 ]} is finite (by the previous point). In particular, this estimates implies that I(t) tends to 0 when t = 0 tends to 0. Since I(0) = 0, the function I is continuous at 0. By symmetry it is also continuous at 1, with I(1) = 0.

Consider the map K : R + → R + defined by K(x) = e x I (e -x ) . Then for all x, y ∈ R + , by [START_REF] Kalai | Threshold phenomena and influence: perspectives from mathematics, computer science, and economics[END_REF] K(x + y) = I (e -x e -y ) e -x e -y ≤ e -x I (e -y ) + e -x I (e -y ) e -x e -y = K(y) + K(x), which means that K is subadditive. Moreover, since I is continuous at 1, K is continuous at 0, with

K(0) = I(1) = 0.
For all a > ǫ > 0, we have by subadditivity K(a+ ǫ) ≤ K(a)+ K(ǫ) and K(a) ≤ K(a-ǫ)+ K(ǫ). Letting ǫ tend to zero, we obtain for all a > 0 lim sup

x→a + K(x) ≤ K(a) ≤ lim inf x→a - K(x).
In words, on (0, +∞), the function K is right-upper-semicontinuous and left-lower-semicontinuous. Since for all t ∈ (0, 1), I(t) = t K(log(1/t)), if follows that on (0, 1) the function I is left-uppersemicontinuous and right-lower-semicontinuous. Note that "left" and "right" were exchanged, since t → log(1/t) is continuous decreasing. The symmetry assumption I(t) = I(1t) allows to exchange once more: so I is also right-upper-semicontinuous and left-lower-semicontinuous on (0, 1). Thus I is continuous on (0, 1), and actually on [0, 1]. Indeed, the continuity at the endpoints has already been established.

Next, let us draw another consequence of the above properties of K. Lemma 10 directly applies and gives that

lim h→0 + K(h) h = sup x>0 K(x) x = sup x>0 e x I(e -x )
x .

Since we assume that I is not identically 0, the above limit, denoted by L, belongs to (0, +∞]. We now translate this convergence in terms of I: using symmetry, for t ∈ (0, 1),

I(t) t = I(1 -t) t = 1 -t t K log 1 1 -t = (1 -t) log 1 1-t t × K log 1 1-t log 1 1-t .
When t > 0 tends to 0, the first ratio tends to 1, and the second to L = lim h→0 + K(h)/h. Therefore we can deduce that lim t→0 + I(t)/t = L ∈ (0, +∞].

In order to turn this limit into a lower bound on I(t)/t for t ∈ (0, 1/2], we need to check that I does not vanish in (0, 1). To do this, let us consider the set S 0 = {x ∈ (0, 1); I(x) = 0}. By [START_REF] Kalai | Threshold phenomena and influence: perspectives from mathematics, computer science, and economics[END_REF], it is stable by product and symmetry around 1/2. If it were non-empty, the first part of Lemma 9 would imply that S 0 is dense in (0, 1). By continuity of I, we would conclude that I is identically 0. Since, we assumed that I does not vanish everywhere, it follows that S 0 = ∅. As a conclusion, the function I vanishes only at 0 and 1.

On (0, 1/2] the map t → I(t)/t is continuous, with positive values. Moreover it has a positive (maybe infinite) limit at 0 + . As a consequence, there exists c > 0 such that I(t) ≥ ct for all t ≤ 1/2.

Let us deduce that I is essentially non-decreasing on [0, 1/2]. Let 0 ≤ s < t ≤ 1 2 . Using the two-points inequality (10)

I(s) = I t × s t ≤ s t I(t) + tI s t ≤ I(t) + t max I ≤ 1 + max I c I(t),
where we have used that I is continuous on [0, 1] and I(t) ≥ ct.

Eventually, let us prove that I J 1 is essentially non-decreasing on (0, 1). Let 0 < s < t < 1. Then one can write s = t k+α with k =

I(s) s = I t k+α t k+α ≤ k I(t) t + I (t α ) t α . (11)
Assume first that t ≥ 1 2 . Then t α ≥ t ≥ 1 2 . We have shown that I is essentially non-decreasing on [0, 1 2 ] (with a constant denoted by D). By symmetry, it follows that I is essentially non-increasing on [ 1 2 , 1] with constant D. Hence

I (t α ) t α ≤ D I (t) t α ≤ D I (t) t .
Combining this estimate with [START_REF] Keller | On the influences of variables on Boolean functions in product spaces[END_REF] gives

I(s) s ≤ (k + D) I(t) t ≤ (1 + D)k I(t) t ≤ (1 + D) log 1 s log 1 t I(t) t ,
that is I(s) J 1 (s) ≤ (1 + D) I(t) J 1 (t) . In particular, we have shown that I/J 1 is essentially non-decreasing on [1/2, 1]. Using the symmetry of I, this implies that on [0, 1/2] the function I(t)/J 0 (t) = I(t)/t is essentially non-increasing. This is actually explained in the first part of the proof of Lemma 5, see Equation [START_REF] Bobkov | Isoperimetric problem for uniform enlargement[END_REF]. We have already shown that I is essentially non-decreasing on (0, 1/2]. Thus by symmetry, I is essentially non-increasing on [1/2, 1], and so is the map t → I(t)/t = I(t)/J 0 (t). Therefore, I/J 0 is essentially non-increasing on the whole interval (0, 1]. Let us denote by D 0 the corresponding constant.

The latter fact allows to conclude: Let 0 < s < t < 1. Since t α ≥ t, we know that

I(t α ) t α ≤ D 0 I(t)
t . Combining this estimate with [START_REF] Keller | On the influences of variables on Boolean functions in product spaces[END_REF] gives,

I(s) s ≤ k I(t) t + I (t α ) t α ≤ (k + D 0 ) I(t) t ≤ (1 + D 0 )k I(t) t ≤ (1 + D 0 ) log 1 s log 1 t I(t) t .
The proof is now complete.

An application to geometric influences

This section is devoted to an application of Theorem 4 to geometric influences. The notion of influence of a variable on a boolean function plays an important role in discrete harmonic analysis, with applications to various fields (see e.g. the survey article [START_REF] Kalai | Threshold phenomena and influence: perspectives from mathematics, computer science, and economics[END_REF] on threshold phenomena). Let us recall the definition: for a function f : {0, 1} n → {0, 1}, which can be viewed as a subset A = {x; f (x) = 1} of {0, 1} n , the influence of the i-th variable with respect to a probability measure ν on the discrete cube {0, 1} n is

I i (f ) = I i (A) := P x∼ν f (x) = f (τ i (x)) = P x∼ν x ∈ A xor τ i (x) ∈ A ,
where τ i (x) is the neighbour of x having different i-th coordinate, (τ i (x)) i = 1x i . Geometrically speaking, I i (A) measures the size of the edge boundary of A in the i-th direction. A seminal result in the theory of influences is the KKL theorem (by Kahn, Kalai and Linial [START_REF] Kalai | The influence of variables on boolean functions[END_REF]). Based on the hypercontractivity inequality, it ensures the existence of a coordinate with a large influence for non-constant boolean functions.

Recent papers by Keller [START_REF] Keller | On the influences of variables on Boolean functions in product spaces[END_REF] and Keller, Mossel and Sen [START_REF] Keller | Geometric influences[END_REF] develop the theory of influences in the case of a continuous space. They propose two different definitions: h-influences [START_REF] Keller | On the influences of variables on Boolean functions in product spaces[END_REF] involve the measures of the intersections of a given set with all lines in the i-th canonical direction, while geometric influences [START_REF] Keller | Geometric influences[END_REF] involve the boundary measures of the intersections with lines in the i-th direction.

Definition. Let n ∈ N * , i ∈ {1, ..., n}, x ∈ R n and A a Borel subset of R n . For z ∈ R n-1 , we set

A z i = y ∈ R (z 1 , ..., z i-1 , y, z i , ..., z n-1 ) ∈ A .
Let ν = ν 1 ⊗ ... ⊗ ν n be a product probability measure on R n . If h : [0, 1] → R + is a measurable function, the h-influence of the i-th coordinate on A with respect to ν is defined by

I h ν,i (A) = R n-1 h (ν i (A z i )) d ν i (z),
where

ν i = ν 1 ⊗ ... ⊗ ν i-1 ⊗ ν i+1 ⊗ ... ⊗ ν n .
The geometric influence of the i-th coordinate on A with respect to the measure ν is given by

I G ν,i (A) = R n-1 (ν i ) + (A z i ) d ν i (z).
When the choice of the underlying measure is obvious, we simply write I h i (A) and I G i (A). Keller was able to prove an analogue of the KKL theorem for h-influences provided h is larger than the entropy function Ent defined by Ent(x) = x log 1

x + (1x) log 1 1-x for x ∈ (0, 1) and Ent(0) = Ent(1) = 0. His result [START_REF] Keller | On the influences of variables on Boolean functions in product spaces[END_REF] is stated for functions on the unit cube, equipped with Lebesgue's measure. Using a standard transportation argument yields the following formulation: Theorem 6. Let µ be a probability measure on R. Then, for every Borel set A ⊆ R n max 1≤i≤n

I Ent i (A) ≥ γ µ n (A)µ n (A c ) log n n ,
where γ > 0 is a universal constant.

Keller, Mossel and Sen [START_REF] Keller | Geometric influences[END_REF] establish an analogue of the KKL theorem for geometric influences for Boltzmann measures dµ ρ (t) = exp(-|t| ρ )dt/Z ρ dt with ρ ≥ 1 (and under mild assumptions for log-concave measures enjoying the same isoperimetric inequality as µ ρ ). Thanks to Theorem 4 we can propose a more general result: Theorem 7. Let µ be an even log-concave probability measure on R, with positive and C 1 -bounded density ϕ µ . Assume that I µ ≥ J where J is a non-negative function on [0, 1], which is symmetric with respect to 1/2, verifies J(0) = 0 and t → J(t)/(t log(1/t)) is essentially non-decreasing on (0, 1) with constant D. Then for every Borel set

A ⊂ R n , max 1≤i≤n I G i (A) ≥ α D µ n (A)µ n (A c )J 1 n ,
where α D ≥ κ D 3 and κ > 0 is a universal constant.

Remark 4. Actually the conclusion of Theorem 7 holds under less restrictive conditions on the measure. In particular the log-concavity assumption can be removed, either by using a different symmetrization argument than in [START_REF] Keller | Geometric influences[END_REF] or by introducing another definition of the geometric influence based on notions of geometric measure theory. These modifications require a substantial and technical work that will appear in the PhD dissertation of the second-named author. In the present paper, we simply explain how the argument of Keller, Mossel and Sen can be adapted, putting forward the parts of the reasoning where the conditions on J are used.

The next lemma follows from Proposition 1.3 in [START_REF] Keller | Geometric influences[END_REF]. It explains the connection between geometric influences and boundary measure for the uniform enlargement: Lemma 11. Let µ be as in Theorem 7. Let A ⊆ R n be a monotone increasing set (in the following sense: if x ∈ A and for all i, x i ≤ y i , then y ∈ A). Then

n i=1 I G i (A) = (µ n ) + (A).
Proof of Theorem 7. We follow the argument of Keller, Mossel and Sen. Assume n ≥ 2. Let A ⊆ R n be as in the statement of the theorem. Lemma 3.7 of [START_REF] Keller | Geometric influences[END_REF] ensures that, without loss of generality, one can assume that A is increasing. Set t = µ n (A). Since A and A c have the same influences, we may assume that t ≤ 1/2 (A c is monotone decreasing, but passing to its image by the symmetry with respect to the origin we may ensure that we work is an increasing set of measure at most 1/2). We distinguish two cases : First case : t ≤ 1/n. Thanks to Lemma 11 and to the isoperimetric inequality of Theorem 4

n i=1 I G i (A) = (µ n ) + (A) ≥ 1 c D J (µ n (A)) = 1 c D J(t),
with c D = 2D 2 (log 2) 2 . Lemma 5 asserts that s → J(s)/s is essentially non-increasing on (0, 1/2] with constant D/ log 2, therefore J(t)/t ≥ nJ(1/n) log 2/D. Consequently

max i I G i (A) ≥ 1 n n i=1 I G i (A) ≥ log 2 Dc D tJ 1 n ≥ (log 2) 3 2D 3 t(1 -t)J 1 n .
Second case : t ∈ (1/n, 1/2]. The argument uses three main ingredients. The first one is the isoperimetric inequality I µ ≥ J, which implies that for all i ≤ n,

I G i (A) = R n-1 µ + (A z i ) dµ n-1 (z) ≥ R n-1 J (µ (A z i )) dµ n-1 (z) = I J i (A). (12)
The second ingredient is a comparison between J-influences and Ent-influences: observe that Ent is symmetric with respect to 1/2 and is increasing and one-to-one on [0, 1/2]. Let Ent -1 : [0, log 2] → [0, 1/2] be its reciprocal function. Then for all i ≤ n,

I J i (A) ≥ J(s) 2D with s := Ent -1 I Ent i (A) 2 ∈ 0, 1 2 . ( 13
)
We postpone the proof of this inequality, and explain how to conclude. The last ingredient is Keller's version of the KKL inequality (Theorem 6). It provides an index i such that I Ent i (A) ≥ γt(1t) log(n)/n, where γ > 0 is a universal constant. Observe for further use that necessarily γ < 8 (indeed, Ent ≤ log 2 and one can choose n = 2 and t = 1/2 in Keller's theorem). Let us show that the i-th coordinate has a large geometric influence.

Observe that for every y ∈ 0, 1 2 ⊂ [0, log 2], θ(y) := y 2 log 1 y ≤ Ent -1 (y). Since I Ent i (A)/2 ≤ log(2)/2 ≤ 1/2, and θ is increasing on (0, 1), , where the latter inequality relies on t(1t) ≥ (1t)/n ≥ 1/(2n). Since γ ≤ 8, the last fraction in the lower bound of s is a positive function of n ≥ 2 with a positive limit when n tends to infinity.

s = Ent -1 I Ent i (A) 2 ≥ θ I Ent i (A) 2 ≥ θ γ 2
Hence there exists c = c(γ) > 0 such that s ≥ ct(1t)/n. It remains to combine this estimate with I G i (A) ≥ J(s)/(2D), a consequence of ( 12) and (13): If s ≤ 1 n , then we also use Lemma 5, which asserts that u → J(u)/u is essentially non-increasing on 0, 1 2 with constant D log 2 :

I G i (A) ≥ J(s) 2D ≥ log 2 2D 2 s J 1 n 1 n ≥ log 2 2D 2 ct(1 -t)J 1 n .
If s > 1 n then, using the fact that J is essentially non-decreasing on 0, 1 2 with constant 2D e log 2

(see Lemma 6), we get

I G i (A) ≥ J(s) 2D ≥ e log 2 4D 2 J 1 n ≥ e log 2 D 2 t(1 -t)J 1 n .
Eventually, we give a proof for [START_REF] Kuczma | An introduction to the theory of functional equations and inequalities[END_REF]. By hypothesis, J/J 1 is essentially non-decreasing with constant D, where J 1 (x) = x log(1/x). Observe that for x ∈ 0, 1 2 , J 1 (x) ≤ Ent(x) = J 1 (x) + J 1 (1x) ≤ 2J 1 (x).

It follows that J

Ent is essentially non-decreasing on 0, 1 2 with constant 2D. Recall that s ∈ 0, Ent (µ (A z i )) dµ n-1 (z) ≥ I Ent i (A) 2 .

Therefore, using in addition the symmetry with respect to 1/2 of J and Ent and the fact that J Ent is essentially non-decreasing on (0, 1/2] with constant 2D, we get

I J i (A) ≥ µ(A z i )∈[s,1-s] J (µ (A z i )) dµ n-1 (z) = µ(A z i )∈[s,1-s] J min(µ (A z i ) , 1 -µ (A z i )) dµ n-1 (z) ≥ 1 2D J(s) Ent(s) µ(A z i )∈[s,1-s] Ent min(µ(A z i ), 1 -µ(A z i )) dµ n-1 (z) = 1 2D J(s) Ent(s) µ(A z i )∈[s,1-s] Ent µ(A z i ) dµ n-1 (z) ≥ 1 2D
J (s) Ent (s)

I Ent i (A) 2 = J (s) 2D .
The proof is complete.

1 / 2 ]

 12 , we choose α = 1 and get sup α∈(0,1]Φ(α) 1y α ≥ (1y)Φ(1)However, y ≥ 1/e ensures 1/ log(1/y) ≥ 1. Since Φ is essentially non-increasing, C 0 Φ(1) ≥ Φ(1/ log(1/y)). Therefore we have proved the first claim, with a constant min(1e -1 , 1/(2C 0 )) = 1/(2C 0 ).To prove the converse inequality, we change variables as follows: setting c = α log(1/y), sup α∈(0,1] Φ(α) 1y α = sup c∈(0,log(1/y)]

log 1 s log 1 t∈

 1 N * and α ∈ [0, 1). By the two-points inequality for I:

1 2verifies

 1 Ent (s) = I Ent i (A)/2. Note that, if x / ∈ [s, 1s], then Ent(x) < I Ent i (A)/2. This yields µ(A z i )∈[s,1-s] Ent (µ (A z i )) dµ n-1 (z) = I Ent i (A)µ(A z i )/ ∈[s,1-s]

* Partially supported by ANR 2011 BS01 007 01 (GeMeCoD project)