Guillaume Dramais

M Jodeau
email: jodeau@lyon.cemagref.fr

A Hauet

A Paquier

J Le Coz

Application and evaluation of LS-PIV technique for the monitoring of river surface velocities in high flow conditions abbreviated title: LS-PIV measurements in high flow conditions

Keywords: LS-PIV, river discharge measurement, flow velocity, high flow condition, 2Dh numerical calculation

Application and evaluation of LS-PIV technique for the monitoring of river surface velocities in high flow conditions

Introduction

The investigation of flow velocities is essential in the study of sediment deposition and erosion patterns in rivers, see for instance [START_REF] Sukhodolov | On the correspondance between morphological and hydrodynamical patterns of groyne fields[END_REF][START_REF] Ashworth | Mid-channel bar growth and its relationship to local flow strength and direction[END_REF][START_REF] Whiting | Experimental studies of bed topography and flow patterns 28 Author-produced version of the article[END_REF]. Especially, the knowledge of flow conditions associated with floods is required for the study of river bed morphology, as high flows mainly contribute to morphological changes (for field example, see [START_REF] Eaton | Effects of large floods on sediment transport and reach morphology in the cobble-bed Sainte Marguerite River[END_REF]).

In mountain streams, velocity measurements and discharge estimates are usually based on leaded current-meter measurements from bridges or wires set across the river width. Usual velocity measurements in rivers using leaded current-meter provide data with poor spatial and temporal resolutions. For instance, one hour is needed to measure velocities over a 50 m cross-section and measurements are often limited to one cross-section in the vicinity of a bridge. Other measurement solutions like Doppler profilers (aDcp) involve intrusion in the river (boat or floating board) which is not always easy and safe [START_REF] Lu | Application of a portable measuring system with acoustic Doppler current profiler to discharge observations in steep rivers[END_REF]. During high flow conditions, high flow velocities, large water depths and large floating objects (trunk, debris) prevented any kind of systematic intrusive flow measurements. Non-intrusive velocity measurements are needed when natural conditions do not permit the use of conventional techniques. Chemical dilution methods can provide precise discharge estimates but they do not give information on the velocity field structure.

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Partly due to the improvement of computing capabilities, digital imaging techniques are increasingly being used in environmental field measurements. In the present study, LS-PIV (Large Scale Particle Image Velocimetry) is used to measure velocity fields at the flow surface without intrusion in the flow at the river scale. LS-PIV is an image based method which consists in recording images of the flow surface and calculating the most likely displacements of patterns using statistical identification [START_REF] Fujita | Large-scale particle image velocimetry for flow analysis in hydraulic engineering applications[END_REF].

This velocity measurement technique is used in laboratory experiments [START_REF] Weitbrecht | Large scale PIV-measurements at the surface of shallow water flows[END_REF][START_REF] Admiraal | Case study: Particle velocimetry in a model of lake Ogallala[END_REF][START_REF] Kantoush | Experimental study of suspended sediment transport and deposition in a rectangular shallow reservoir[END_REF], as well as in natural conditions [START_REF] Creutin | River gauging using PIV techniques: a proof of concept experiment on the Iowa river[END_REF]. Unlike PIV used in laboratory with laser illumination that allows the exploration of the whole flow, LS-PIV with natural light is restricted to surface flow investigation. LS-PIV method has been used in high flow conditions to measure flow velocities [START_REF] Fujita | Large-scale particle image velocimetry for flow analysis in hydraulic engineering applications[END_REF], and up to now, this technique has been used in rivers for many applications: river discharge calculation [11,[START_REF] Hauet | Experimental system for real-time discharge estimation using an image-based method[END_REF], aquatic habitat mapping studies [START_REF] Smith | A field application of particle image velocimetry (PIV) for the measurement of surface flow velocities in aquatic habitat studies[END_REF], assessment of mean flow patterns between groins [START_REF] Fujita | Image-based technique for river monitoring[END_REF]... Natural floods often carry many objects or turbulence-driven patterns (ripples, boils) which are more or less visible at the surface of the flow. If dense enough, they can be used as traceable patterns in the LS-PIV procedure, for instance floating ice can be considered [START_REF] Jasek | Estimation of Yukon River discharge during an ice jam near Dawson City[END_REF]. Artificial seeding is required when few natural tracers are visible at the surface of the flow [START_REF] Fujita | Unseeded and seeded PIV measurements of river flows videotaped from a helicopter[END_REF][START_REF] Admiraal | Case study: Particle velocimetry in a model of lake Ogallala[END_REF][START_REF] Muto | Velocity measurements in open channel flow with rectangular embayments formed by spur dikes[END_REF][START_REF] Kantoush | Experimental study of suspended sediment transport and deposition in a rectangular shallow reservoir[END_REF].

In this paper, LS-PIV is applied in field conditions during a reservoir release. These velocity measurements are part of an extensive investigation of the morphological changes over a gravel alternate bar [START_REF] Jodeau | Effect of a 30 Author-produced version of the article published Flow Measurement and Instrumentation[END_REF].

In order to estimate flow velocities over the bar during the high flow event, a complete installation was elaborated to record large images of the flow which can be analyzed using 3 LS-PIV method according to [START_REF] Hauet | Estimation de débit et mesure de vitesse en rivière par Large-Scale Particle Image Velocimetry, Discharge estimates and velocity measurements using Large-Scale Particle Image Velocimetry[END_REF][START_REF] Hauet | Experimental system for real-time discharge estimation using an image-based method[END_REF]. The installation includes (i) a remotely controlled mobile image recording system from an elevated point and (ii) an artificial flow seeding system as no natural tracers were visible at the surface of the flow. To authors' knowledge this kind of installation has not been used in rivers before. The LS-PIV method was improved to consider the specific constraints of these high flow conditions: significant variations of the water surface elevation, dark water color due to very high fine sediment loads, tree shadows and sun reflections on the flow surface. Evaluation of LS-PIV technique adapted to high flow condition and mountain stream is performed owing to comparisons with 2Dh numerical calculations. Flow discharge is computed from LS-PIV measurements throughout the whole event. Direct comparison is made with two leaded current-meter measurements and with the discharge calculated by river managers from hydraulic structure laws.

The first part of this paper presents the measurement installation. The second part describes the LS-PIV method. The third part details flow velocity patterns and comparison between experimental results and 2Dh numerical calculations. The fourth part deals with discharge calculation and comparison with discharge estimates from leaded current-meter measurements.

Measurement installation

Mast and video camera

A commercially available digital video camera (Canon MV 750i) was used to record the river flow. Sequences format was mini-DV at a rate of 25 frames per second and image resolution was 540×720 pixels. The video camera was set on a mobile lightweight telescopic mast which height can be set from 2 to 10 m (Maxi Primo, Fireco Components). Four shrouds were hold 4 from the top of the mast to the ground to reduce wind disturbance (Figure 1 b). The camera was remotely controlled from the ground to set the camera on and to adjust view angles.

The movies were directly saved on a laptop. Several Ground Reference Points (GRPs) were positioned in the view field at the same elevation using squares painted on embankments and white square targets to allow further geometric correction of the pictures.

Artificial tracer feeding device

As the surface flow did not show clearly visible patterns, a simple feeding device was developed to supply the flow with Ecofoam chips, a water soluble, foamed filling material created from cornstarch. It is biologically degradable, and as supplied quantities are insignificant in regards to the water flux, it does not affect the water quality of the river. As high flow conditions prevent any sailing or wading in the river, a 50 m-long rope was set up between both banks 50 m upstream from the camera view-point. A sliding container made of a plastic bag disseminated white chips over the flow (Figure 1). The installation is quite similar to a tyrolean traverse. The disseminator movement was controlled from both banks by operators pulling a rope fastened to the chip container. At first, the tracers float on the surface, then they dissolve in water within less than one minute.

During the 1 day-long dam flushing operation, eight sequences of chip fed flows (and 17 sequences without artificial tracers) were recorded over a wide range of discharges and analyzed using LS-PIV method.

The LS-PIV analysis was conducted on around 200 to 1500 image pairs per sequence, with a 0.2 s interval time between images.

Improved LS-PIV technique

Geometric rectification: GRP and variations of water surface elevation

The first step in the LS-PIV procedure consists in geometrically correcting images to obtain ortho-images (Figure 2). This is usually realized by considering the water surface as a plan of constant elevation, which leads to an 8-parameters plan-to-plan perspective projection that can be solved using at least 4 GRPs located at the river free-surface elevation (see [START_REF] Fujita | Visualization and PIV measurement of river surface flow[END_REF] for instance). In this study, the water level varied for more than one meter during the dam release. We placed the GRPs in a horizontal plan parallel to the river surface which elevation was higher than the expected maximum water level during the flushing event. The conventional projection equations corrected to take into account the water level changes are

i = a 1 x + a 2 y + a 3 z + a 4 c 1 x + c 2 y + c 3 z + 1 (1)
j = b 1 x + b 2 y + b 3 z + b 4 c 1 x + c 2 y + c 3 z + 1 (2)
where [i, j] are the coordinates in the image coordinate system (in pixel) of the point of Cartesian coordinates [x, y, z]. As the river surface can be considered planar, the coordinate z can be expressed as:

z = d 1 x + d 2 y + d 3 (3)
Using the assumption that the river surface slope is small at the considered scale of study, we reduce the previous equation to z = d 3 (t). Combining the expression of z in Equation

(1) leads to:

i = a 1 x + a 2 y + a 3 d 3 (t) + a 4 c 1 x + c 2 y + c 3 d 3 (t) + 1 (4) j = b 1 x + b 2 y + b 3 d 3 (t) + b 4 c 1 x + c 2 y + c 3 d 3 (t) + 1 (5)
where d 3 (t) is simply the difference of level between the plan of the GRPs and the water level given by the limnigraphs at time t. The system of Equation ( 4) and ( 5) has 11 unknowns, so it can be solved using at least N ≥ 6 GRPs giving 13 coordinates: z GRP and (x i ,y i ) where i=1 to 6 as:

EF = G where E = [11 × 2N ] =                          x 1 y 1 d 3 1 0 0 0 0 -i 1 x 1 -i 1 y 1 -i 1 d 3 x 2 y 2 d 3 1 0 0 0 0 -i 2 x 2 -i 2 y 2 -i 2 d 3 . . . . . . . . . . . . . . . . . . . . . . . . x N y N d 3 1 0 0 0 0 -i N x N -i N y N -i N d 3 0 0 0 0 x 1 y 1 d 3 1 -j 1 x 1 -j 1 y 1 -j 1 d 3 0 0 0 0 x 2 y 2 d 3 1 -j 2 x 2 -j 2 y 2 -j 2 d 3 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 x N y N d 3 1 -j N x N -j N y N -j N d 3                          (6)
F = [11 × 1] = [a 1 , a 2 , a 3 , a 4 , b 1 , b 2 , b 3 , b 4 , c 1 , c 2 , c 3 ] T (7) G = [2N × 1] = [i 1 , i 2 , . . . , i N , j 1 , j 2 , . . . , j N ] T (8)
The equation is solved using a least-square method:

F = (E T E) -1 E T G (9)
The inverse relation mapping image coordinates onto Cartesian coordinates is:

x = m 1 i + m 2 j + m 3 p 1 i + p 2 j + p 3 (10)
y = q 1 i + q 2 j + q 3 p 1 i + p 2 j + p 3 (11)
where the m i , p i and q i parameters can be found in Annex 1.

LS-PIV data processing

A classical cross-correlation algorithm was used to determine the displacement of the flow tracers. In this study, we use a PIV algorithm for large scale applications with low resolution images, developed by [START_REF] Fincham | Low cost, high resolution DPIV for measurement of turbulent fluid flow[END_REF]. It calculates the correlation between the interrogation area (IA) centered on a point a ij in the first image (image A) and the IA centered at point b ij in the second image (image B) recorded with a time interval of δt seconds. The correlation coefficient R(a ij , b ij ) is a similarity index for the gray-scale intensity of a group of pixels contained in the two compared IAs, expressed as:

R(a ij , b ij ) = 1≤i≤M i 1≤j≤M j A ij -A ij B ij -B ij 1≤i≤M i 1≤j≤M j A ij -A ij 2 1≤i≤M i 1≤j≤M j B ij -B ij 2 1/2 (12)
where M i and M j are the sizes of the interrogation areas (in pixels), and A ij and B ij are the distributions of the gray-level intensities in the two interrogation areas. Correlation coefficients are only computed for points b ij within some search area (SA).

The PIV approach assumes that the most probable displacement of the fluid from point a ij during period δt is the one corresponding to the maximum correlation coefficient. Sub-pixel displacement accuracy can be reached using a parabolic fit [START_REF] Forliti | Bias and precision errors of digital particle image velocimetry[END_REF]. Velocity vectors are derived from these displacements by dividing them by δt. The process is iteratively conducted for the whole image. The interrogation area was set as covering a physical surface of 2×2 m 2 , which is small enough to preserve the scale of interest in the flow since any flow scales smaller than the size of the interrogation area are lost through processing, and large enough to include recognizable tracer patterns within it, i.e. to encompass one or more typical chips patterns that are used to trace the flow free surface.

LS-PIV analysis gave the instantaneous and time-averaged 2D surface velocity fields. Due to some temporary lacks of clearly visible patterns in some limited areas of the view field, it happens that instantaneous surface velocity is not estimated in such areas, they are com- [START_REF] Admiraal | Case study: Particle velocimetry in a model of lake Ogallala[END_REF][START_REF] Le Coz | Recirculating flow assessment from aDcp, LS-PIV and 2Dh modelling[END_REF][START_REF] Meselhe | Large scale image velocimetry for low velocity and shallow water flows[END_REF]. In the present work LS-PIV performances are evaluated without implementing any systematic criteria on calculated velocities.

Intensity threshold: tracers and stationary waves

The flow regime during the high discharge period was highly turbulent, locally creating hydraulic jumps and stationary waves. In many LS-PIV post-processing methods all the visible patterns are taken into account [START_REF] Creutin | River gauging using PIV techniques: a proof of concept experiment on the Iowa river[END_REF][START_REF] Fujita | Visualization and PIV measurement of river surface flow[END_REF]. The specular reflection of the sun on those waves created bright quasi-fixed patterns, but not immobile like trees for instance. A threshold on gray-scale images was used to focus on chip patterns (that appear completely white on the images) and to avoid taking into account light reflection on stationary waves (that appear darker). The threshold value is related to the image global illumination level.

Some attempts to analyze images without tracers led to a significant underestimation of surface flow velocities. Indeed comparison at the same calculation node between velocity estimates using a threshold (v1) and velocity estimates using classical calculation (v2) (Fig- The work of [START_REF] Meselhe | Large scale image velocimetry for low velocity and shallow water flows[END_REF] suggests that a seeding-density 10-30% of the surface may be enough to avoid major error in the estimation of surface velocities. In this field application, as tracer patterns were restricted to artificial seeding, tracer density is less than 10% of the river flow surface. It leads to large areas without calculated velocities, but this effect is corrected using a time-average processing (3.2) adapted to low tracer density.

Artificial seeding permits to add tracers in the far field. Without artificial tracers, velocity vectors were only obtained in the near field region because no image patterns were captured in the far-field. This effect of the decreasing of resolution over the image was highlighted by [START_REF] Fujita | Refinement of LSPIV technique for monitoring river surface flows[END_REF]. In the present field study, the white color of tracers is essential because it maximizes the contrast with the black color of the river. However, in shadow areas, contrast between water and tracers is poorer than in the rest of the image leading to flow areas without velocity assessment.

Evaluation of LS-PIV surface velocity measurements

The installation and method described above were evaluated on two different flow configurations in a mountain stream during a reservoir release.

Site description and flushing operation

The field work took place on the River Arc, France, during a reservoir release (June, 27 2006). Once a year this gravel bed river undergoes a reservoir release to preserve hydropower capacity. During flushing operations, discharge varied approximately from 10 m 3 /s to 150 m 3 /s and water color was very turbid due to very high concentrations of fine suspended sediments (between 10 and 30 g/l). Two distinct study sites were selected on the same ap- Additional measurements were performed during the flushing event: water level was measured at four points using bubble pipe devices (ISCO 4230, one measurement per minute), two discharge measurements with leaded current meter were performed at a bridge 200 m downstream. Low flow discharge was regularly monitored over the year using wadable currentmeter measurements. the bathymetry of the river reach was surveyed before and after the flushing event with a GPS system (Trimble 5800 RTK).

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Relationship between surface velocity and depth-averaged velocity

In order to compare LS-PIV measurements with conventional measurements or with depthaveraged numerical results, the relationship between surface velocity and depth-averaged velocity must be established. As considered in many previous studies [START_REF] Rantz | Measurement and computation of streamflow[END_REF][START_REF] Wilcock | Specification of sediment maintenance flows for a large gravel-bed river[END_REF][START_REF] Smart | Turbulent velocity profiles and boundary shear stress in gravel bed rivers[END_REF][START_REF] Je | Use of radar to monitor stream discharge by noncontact methods[END_REF] tical velocity profiles are assumed to follow a logarithmic law : v(z) = A ln( z z 0 ), where A is a coefficient, z the elevation above the bed and z 0 the elevation at which velocity is zero, depending on the bed roughness. This vertical generic law was confirmed by velocity measurements performed at the bridge at high discharge (102 m 3 /s and 115 m 3 /s) and detailed vertical velocity profiles performed at low flow (5.5, 16 and 20 m 3 /s). Dimensionless data fit reasonably with dimensionless logarithmic velocity profiles whatever the intensity of the discharge is (Figure 4). The bed roughness parameter z 0 can be deduced from fitted log law, leading to z 0 values from 0.001 m to 0.015 m, which is coherent with observed surface grain-sizes of several centimeters. The float coefficient α v = v av /v surf ace can be deduced from the dimensionless logarithmic law: w(z/h = 1) = v(z = h)/v av = 1/α v . This simple relationship between surface and depth-averaged velocities (

α v ≈ 1-1/ln(h/z 0 ) ≈ constant)
is valid only for h > 100z 0 , which is equivalent to approximately 40 cm. Data presented here lead to α v from 0.75 to 0.83, for this data set the condition h > 100z 0 is always fulfilled. In the following calculations, the mean value of all the experimental values, α v = 0.79, will be used to convert surface velocity into depth-averaged velocity. This value is smaller than the usual 0.85 coefficient used for instance by [START_REF] Rantz | Measurement and computation of streamflow[END_REF][START_REF] Creutin | River gauging using PIV techniques: a proof of concept experiment on the Iowa river[END_REF].

Comparison with 2Dh calculations

The depth-averaged (2Dh) water flow in the reach was modeled with the two-dimensional model Rubar20. The code Rubar20 uses a finite volume scheme that is explicit, Godunovtype and second order accurate [START_REF] Paquier | Modélisation et simulation de la propagation de l'onde de rupture de barrage[END_REF]. It solves the 2-dimensional shallow water equations written below :

∂h ∂t + ∂hu ∂x + ∂hv ∂y = 0 (13) ∂hu ∂t + ∂(hu 2 + gh 2 /2) ∂x + ∂huv ∂y = -gh ∂Z ∂x -gn 2 M u √ u 2 + v 2 h 1/3 + K ∂ ∂x h ∂u ∂x + ∂ ∂y h ∂u ∂y ( 14
)
∂hv ∂t + ∂huv ∂x + ∂(hv 2 + gh 2 /2) ∂y = -gh ∂Z ∂y -gn 2 M v √ u 2 + v 2 h 1/3 + K ∂ ∂x h ∂v ∂x + ∂ ∂y h ∂v ∂y (15)                       
in which u and v are depth-averaged velocities along respectively x and y axis, h water depth, Z bottom level, g gravity acceleration, n M Manning coefficient, K the viscosity coefficient.

The mesh is constituted of quadrangles or triangles which have 0 or 1 common edge, it includes the whole gravel bar (Figure 5). Bed topographies before and after the event were meshed. In order to obtain the mean water level along the reach of interest, the Manning friction coefficient n M is taken constant over the whole reach (n M =0.025 m -1/3 s), excepted on boulder embankments (n M =0.05 m -1/3 s) to reproduce higher friction due to large boulders and sparse vegetation and at the bar tail (downstream part) (n M =0.017 m -1/3 s) to account for finer gravels. At the upstream cross-section, we used a hydrograph and water level measurements as boundary conditions. The hydrograph is the discharge time series presented in 4.1 (Figure 8). At the downstream cross-section the water level is set to critical regime.

Difficulties to fit the simulated water level line to measurements were encountered mainly because water level lines from calculation are significantly influenced by the topography used to build the mesh. This numerical approach does not take into account gravel bed changes.

Our topography measurements show obvious changes of the bed during the flushing event.

Knowing this, it is quite difficult to choose which topography corresponds to a given period of time. Figure 6 shows comparison between LS-PIV measurements and velocities calculated by the model. As the LS-PIV calculation grid is finer than the numerical model mesh, each of the numerical calculations is compared to the nearest LS-PIV assessment. Two conditions are applied to preserve the relevance of the comparison: (i) the compared data must be located nearer than one meter and (ii) the angle between both velocity vectors must be lower than π/5, otherwise it probably means that LS-PIV measurement is biased, for instance due to spurious movements. The float coefficient α v =0.79 was used to compare LS-PIV surface velocities to depth-averaged calculated velocities (see 4.2).

In both sites, angles between numerically calculated velocities and field measurements are

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vectors is mainly less than ±π/40 ≈ 5 • .

On Site D, accordance between calculated and measured velocity intensities is excellent for water depths higher than 50 cm (Figure 6 bD). Discrepancy is observed for low water-depths corresponding to the near field, ie the part of the flow near the left embankment. Visual controls on movies confirm the value of the velocities in the near field, in this area tracers move approximately at the same velocity as tracers in the middle of the river. As the bias seems to depend on the water depth, the underestimation may be linked to inaccurate calculations at low water depths due an erroneous statement of friction. The bias is strongly noticeable for sequence D1 (lowest discharge) in which water-depths are lower than 50 cm in the whole cross-section. As this sequence was monitored at the beginning of the flushing event when discharge varied quickly, a small error in time in the calculated discharge time series could generate a considerable discrepancy between real discharge and estimated discharge. D1 measurements are in better agreement with calculations performed ten minutes later.

On Site C, where the flow is more complex, LS-PIV velocities differ from calculated velocities for a great part (Figure 6 bC). Agreement between both velocities is acceptable in the near field upstream from the connecting channel.

For further comparisons, calculated velocity vectors were interpolated on the cross sections mapped on Figure 2 and LS-PIV surface velocities were multiplied by the float coefficient embankment occurred in some of the sequences due to tree shadows and absence of tracers.

α v =0.79. On Site D,
Considering previous cross-section profiles, it is acceptable to consider constant interpolation in this region. Thus the model validates LS-PIV velocity assessments and could be a way to face lack of field measurement in space and time.

On Site C, Figures 7 (c) and (d) show that the available measurements agree with the calculations. Despite very few measurements in the main channel, the global shape of the transverse profile is similar in both cases.

Discharge calculations

Using low water wading current-meter discharge measurement and both high flow currentmeter discharge measurements at the bridge, an exponential rating curve was built associated

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to n t = 100 equally spaced points on the section (∆l = 0.77 m between two calculation points). A mean longitudinal bed slope of 0.3% deduced from DEMs is applied to extrapolate flow depths from measurements. In case of an absence of measurement due to shadows or absence of tracers on the right side of the stream, extrapolation of surface velocities with constant value, v extr is performed. The number of extrapolated data related to the number of calculation points (100), f extr , gives an estimate of the reliability of the discharge assessment.

The extrapolated values were chosen to be constant, either the average of the measured values for h = 0 or the maximum of cross-section surface velocity values. This second value is coherent with the comparison between measured and numerically calculated velocities (4.3).

Discharges are calculated using the following formula:

Q = nt i=1 v P i h i ∆l = ∆l (1-fextr)nt i=1 v P i h i measurements + ∆l v extr nt i=(1-fextr)nt h i extrapolation (16)
where v P i is the interpolated depth-averaged projected velocity, normal velocity vector to the cross-section, and h i is the interpolated water-depth at point i on the cross section, and ∆l is the distance between two calculation points, n t is the number of calculation points.

Discharge assessments from these calculations are reported in Table 1. Discharge calculations based on LS-PIV assessments without threshold processing are not presented here but give discharge estimates 70 % lower than those presented here. It confirms the need to clearly focus on moving surface patterns.

The measurements performed from the downstream bridge give discharge peak values significantly lower than those estimated by dam managers (120 m 3 /s versus 150 m 3 /s). There is no loss of water between the downstream dam and the study site and one dimensional modeling of the flushing event from the dam to the study site showed no decreasing of the peak value along the river. Even though the uncertainty of discharge measurements with leaded current-meter can be estimated to be ±5% of the true value [START_REF] Sauer | Determination of error in individual discharge measurements[END_REF], this measurement is more reliable than rating-based discharge assessment. Consequently measurements performed with the current meter at the bridge were considered as references for peak discharge values rather than the assessments at the upstream dam.

LS-PIV measurements are compared with the discharge time series built independently as described in 4.1.The resulting discharge time series is plotted on Figure 8, which shows that the shape of the assessed hydrograph proposed by dam managers is similar to the one calculated here whereas peak discharge seems to be overestimated by dam managers.

LS-PIV discharge estimations fit well with both discharge measurements from the bridge and the new computed discharge curve and consequently validate the calculations. Checking assessed data is a major issue to further use of the hydrograph, for instance in numerical modeling of the event or in analysis of morphological changes of the river bed.

During high flow conditions, LS-PIV technique can be used to measure flow surface velocities.

A complete installation including a video acquisition system from a mobile elevated viewpoint and an artificial feeding device similar was used to record large images of a river flow surface during a reservoir release.

Improvement of LS-PIV method has been developed to obtain velocity assessments despite water surface elevation variations, stationary waves and light reflections. The improved LS-PIV method is evaluated for velocity measurements in a moutain stream during a high flow event.

A float coefficient α v =0.79 based on field measurements was used to convert surface velocities into depth-averaged velocities. Comparisons with 2Dh numerical calculations showed good agreement between measured and calculated velocities, thus it is a useful tool to verify and extrapolate LS-PIV data.

Discharge calculation from LS-PIV surface velocities were consistent with current-meter measurements, and validate the calculation of a new discharge time-series which is significantly different in intensity from dam manager assessments.

The installation and methods are now operational for further studies. The facility is not limited to artificial high flow events in rivers and could be applied to the study of natural unpredictable floods, for instance flash-floods occurring in Mediterranean regions, or the monitoring of flow patterns in reservoirs to increase the efficiency of reservoir flushing procedures.

be improved by using several seeding containers or a more sophisticated device, maybe a permanent installation using a wire line instead of a rope would facilitate the seeding. Our results were also affected by shadow limitations, but some improvement on data processing could probably enhance the quality of the results.

Figure 1 :

 1 Figure 1: (a) Schematic view of the LS-PIV installation: mast, video-camera and view field, Ground

6

 6

Figure 2 :

 2 Figure 2: Typical flow images before and after rectification, LS-PIV time-averaged surface velocity

Figure 3 :

 3 Figure 3: Comparison of calculated velocities with (v1) or without (v2) the use of a threshold level

Figure 4 :

 4 Figure 4: 24 dimensionless vertical velocity profiles fitted on 24 velocity data sets measured at high discharge (102 m 3 /s and 115 m 3 /s) and at low flow (5.5, 16 and 20 m 3 /s).

Figure 5 :

 5 Figure 5: LS-PIV installation with the positions of the camera and the view fields, the GRPs and

Figure 6 :

 6 Figure 6: Comparison between LS-PIV velocity measurements (v piv ) and 2Dh numerical calcu-

Figure 7 (

 7 a) and (b) shows that calculations fit fairly well with measurements except in a limited area near the left embankment, as observed in previous comparisons. In this latter area, calculations underestimate velocity intensities (Figure 7 b). The asymmetry of transverse velocity profile is obvious on calculation results whereas measurements show an almost symmetric transverse profile. Lack of data close to the right 20 Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

Figure 7 :

 7 Figure 7: Comparison on cross-section between LS-PIV velocity measurements (v piv ) and 2Dh numerical calculations (v calc ). On Site D (a) D2, in the beginning of the flushing event, Q=57 m 3 /s, and (b) D6 at the peak discharge, Q=122 m 3 /s. On Site C (c) C1, in the rising phase, Q=71 m 3 /s and (d) C2 at the peak discharge, Q=126 m 3 /s.

  with continuous water level monitoring near Site D. As the depth data quickly fluctuate, a 3 minute moving average was used to smooth the water depth signal. The resulting reference discharge time series (Ref) is plotted on Figure 8, fluctuations are within the associated ±5 % curves. Local discharge estimates over the whole flushing event were performed using LS-PIV velocity measurements. Calculation is performed on a simple quasi-transverse oblique stream section downstream to the gravel bar at site D (Figure 2 (b)). Topography of the gravel bed evolved during the flushing event. As only two bed surface DEMs (Digital Elevation Models) are available, the first one standing for topography before the flushing event is used till peak discharge is reached, whereas the second one standing for the topography after the event is used as reference for the second part of the flushing event. Surface velocity values are converted into depth-averaged velocities using a coefficient α v =0.79 (4.2). DEM and depthaveraged projected velocities are computed using inverse distance weighting interpolation 22 Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

Figure 8 :

 8 Figure 8: Discharges from different calculations: assessment from dam managers using incoming

7

  Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

	expressed as:

8

  Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

10

  Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004 puted as zero surface velocity. In order not to underestimate time-averaged velocity, for each calculation node the average is performed only with times leading to non-zero velocity. In many PIV studies several criteria, global or local filtering processes, are used to reduce the number of bad vectors

  Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

	ure 3) shows that v1 is higher than v2 in most of the cases. Points at 0 m/s correspond to
	velocities detected only by one of both analyses. Surface velocities are underestimated by
	0.7 m/s on average for a majority of sequences (D1, D3 to D6). Unfortunately, no obvious
	relationship can be established between analyses without a threshold and analyses with a
	threshold. Thus in the following, only the results obtained from thresholded images were
	used.
	11

12

  Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

  Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

], ver-14

15

  Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

21

  Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

24

  Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

					ESTIMATED DISCHARGE
	sequence							fextr
		time	rating curve (ref)	extrapolation with averaged vel.	extrapolation with maximum vel.
	name							(%)
			(m 3 /s)	(m 3 /s)	diff ref (%)	(m 3 /s)	diff ref (%)
	D1	08:02	17	8	53	9	47	4
	D2	09:01	57	35	39	44	23	31
	D3	09:48	77	81	-5	88	14	32
	D4	09:55	77	77	0	87	13	35
	D5	14:24	119	112	6	121	2	27
	D6	15:39	122	117	4	120	2	10

Table 1 :

 1 Comparison between discharges estimated with the rating curve and discharges calculated with LS-PIV estimates according two methods: extrapolation with width-averaged velocity or extrapolation with maximum velocity.

Author-produced version of the article published Flow Measurement and Instrumentation (2008), 19, 2, 117-127, The original publication is available at http://www.sciencedirect.com DOI: 10.1016/j.flowmeasinst.2007.11.004

Acknowledgments

This study was supported by the Cemagref. Diren Rhône-Alpes performed discharge measurements at the downstream bridge. EDF Groupe d'Exploitation Hydraulique Maurienne provided discharge assessments and authorized measurements during the reservoir release.

Authors are grateful to people who took part in the field survey, in particular F. Thollet and J. Hervé.

Annex 1

Coefficients m i , p i and q i for the inverse relation mapping image coordinates onto Cartesian coordinates read as follows:


Matthieu Fradelizi
email: matthieu.fradelizi@u-pem.fr

Arnaud Marsiglietti
email: arnaud.marsiglietti@u-pem.fr

On the analogue of the concavity of entropy power in the Brunn-Minkowski theory

Keywords: entropy power, parallel volume, parallel set, isoperimetric inequality, Brunn-Minkowski. Mathematics Subject Classification: 52A38, 52A40, 94A17, 53A05

Elaborating on the similarity between the entropy power inequality and the Brunn-Minkowski inequality, Costa and Cover conjectured the 1 n -concavity of the outer parallel volume of measurable sets as an analogue of the concavity of entropy power. We investigate this conjecture and study its relationship with geometric inequalities.

Introduction

First, let us explain the origin of the conjecture of Costa and Cover. Costa and Cover [START_REF] Costa | On the similarity of the entropy power inequality and the Brunn-Minkowski inequality[END_REF] noticed the similarity between the entropy power and the Brunn-Minkowski inequalities: for every independent random vectors X, Y in R n , with finite entropy and for every compact sets A and B in R n one has

N (X + Y ) ≥ N (X) + N (Y ) and |A + B| 1 n ≥ |A| 1 n + |B| 1 n ,
where | • | denotes the n-dimensional Lebesgue measure and

N (X) = 1 2πe e 2 n H(X)
denotes the entropy power of X. Recall that for X with density f the entropy of X is H(X) = -f ln f if the integral exists and H(X) = -∞ otherwise.

Applying the Brunn-Minkowski inequality to B = εB n 2 and letting ε tend to 0 one gets the classical isoperimetric inequality

|∂A| |A| n-1 n ≥ n|B n 2 | 1 n = |∂B n 2 | |B n 2 | n-1 n
, where the outer Minkowski surface area is defined by

|∂A| = lim ε→0 + |A + εB n 2 | -|A| ε ,
whenever the limit exists. In the same way, Costa and Cover applied the entropy power inequality to Y = √ εG, where G is a standard Gaussian random vector (the √ ε comes from the homogeneity of entropy power N ( √ εX) = εN (X)). Then by letting ε tending to 0 and using de Bruijn's identity

d dt H(X + √ tG) = 1 2 I(X + √ tG),
which states that the Fisher information (denoted by I) is the derivative of the entropy along the heat semi-group, they obtained the following "isoperimetric inequality for entropy" N (X)I(X) ≥ n.

Notice that this inequality is equivalent to the Log-Sobolev inequality for the Gaussian measure, see [START_REF] Ané | Sur les inégalités de Sobolev logarithmiques[END_REF] chapter 9. This analogy between the results of the Information theory and the Brunn-Minkowski theory was later extended and further explained and unified through Young's inequality by Dembo [START_REF] Dembo | Information inequalities and uncertainty principles[END_REF] and later on by Dembo, Cover and Thomas [START_REF] Cover | Information theoretic inequalities[END_REF]. Then, Szarek and Voiculescu [START_REF] Szarek | Shannon's entropy power inequality via restricted Minkowski sums[END_REF] deduced the entropy power inequality from a restricted Brunn-Minkowski inequality. Each of these theories deal with a fundamental inequality, the Brunn-Minkowski inequality for the Brunn-Minkowski theory and the entropy power inequality for the Information theory. The objects of each theories are fellows: to the compact sets in the Brunn-Minkowski theory correspond the random vectors in the Information theory, the Gaussian random vectors play the same role as the Euclidean balls, the entropy power N corresponds to the 1/n power of the volume | • | 1/n and, taking logarithms, the entropy H is the analogue of the logarithm of the volume log |•|. Hence one can conjecture that properties of one theory fit into the other theory.

Thus, Costa and Cover [START_REF] Costa | On the similarity of the entropy power inequality and the Brunn-Minkowski inequality[END_REF], as an analogue of the concavity of entropy power with added Gaussian noise, which states that

t → N (X + √ tG)
is a concave function (see [START_REF] Costa | A new entropy power inequality[END_REF] and [START_REF] Villani | A short proof of the "concavity of entropy power[END_REF]), formulated the following conjecture.

Conjecture 1.1 (Costa-Cover [START_REF] Costa | On the similarity of the entropy power inequality and the Brunn-Minkowski inequality[END_REF]). Let A be a bounded measurable set in

R n then the function t → |A + tB n 2 | 1 n is concave on R + .
They also showed using the Brunn-Minkowski inequality that this conjecture holds true if A is a convex set.

Notice that Guleryuz, Lutwak, Yang and Zhang [START_REF] Guleryuz | Information-theoretic inequalities for contoured probability distributions[END_REF] also pursued these analogies between the two theories and more recently, Bobkov and Madiman [START_REF] Bobkov | Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures[END_REF] established an analogue in Information theory of the Milman's reverse Brunn-Minkowski inequality.

In this paper, we investigate Conjecture 1.1 and study its relationship with known geometric inequalities. We prove that the conjecture holds true in dimension 1 for all measurable sets and in dimension 2 for connected sets. In dimension n ≥ 3, we establish that the connectivity hypothesis is not enough and that the conjecture is false in general. We then discuss additional hypotheses which ensure its validity: we conjecture that it holds true for sufficiently large t and we establish it for special sets A. More precisely, our main results are contained in the following theorem.

Theorem 1.2. Let n ≥ 1, A be a bounded measurable set. Define V A (t) = |A + tB n 2 |, t ≥ 0. 1. For n = 1, the function V A is concave on R + . 2. For n = 2, if A is connected then V 1 2
A is concave on R + . Moreover there exists A not connected such that V

1 2
A is not concave on R + .

For

n ≥ 3, if the function ε → |εA + B n
2 | is twice continuously differentiable in the neighborhood of 0, then there exists t 0 such that V

1 n A is concave on [t 0 , +∞). Moreover there exists A connected such that V 1 n A is not concave on R + .
In the next section, we first explain some notations, then we establish analytical properties of the parallel volume and we explore relationships between this conjecture and known geometric inequalities. In the third section, we study the 1 n -concavity property of the parallel volume. In the last section, we investigate further analogies between the Information theory and the Brunn-Minkowski theory.

2 Basic properties of the parallel volume and links with geometric inequalities

Notations

We work in the Euclidean space R n , n ≥ 1, equipped with the ℓ n 2 norm | • |, whose closed unit ball is denoted by B n 2 and canonical basis is {e 1 , . . . , e n }. We also denote by | • | the Lebesgue measure in R n . For non-empty sets A, B in R n we define their Minkowski sum

A + B = {a + b : a ∈ A, b ∈ B}.
We denote by int(A), A, ∂A, conv(A) respectively the interior, the closure, the boundary, the convex hull of the set

A. A function f : R n → R + is 1 n -concave if f 1 n is concave on its support. A set B is a convex body if B is a compact convex set of R n with non- empty interior. If 0 is in the interior of B, then the gauge associated to B is the function • B defined by x B = inf{t > 0 : x ∈ tB}, for every x ∈ R n . Let A be a bounded measurable subset of R n . For x ∈ R n , we set d B (x, A) = inf{ x-y B : y ∈ A} and we simply denote d(x, A) = d B n 2 (x, A). We denote by V A,B the function defined for t ≥ 0 by V A,B (t) = |A + tB|.
When A is convex, V A,B has a polynomial expansion according to the classical Steiner formula

V A,B (t) = |A + tB| = n i=0 t i n i V (A[n -i], B[i]), where V (A[n -i], B[i]
) are called the mixed volume of A, B (see [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory, encyclopedia of mathematics and its applications 44[END_REF] Chapter 5). For B = B n 2 , we simply denote

V A = V A,B n 2 the (outer) parallel volume function of A defined on R + by V A (t) = |A + tB n 2 |.
The outer Minkowski surface area |∂A| of A may be defined using V A : if the function V A admits a right derivative at 0 then one has

(V A ) ′ + (0) = lim t→0 + |A + tB n 2 | -|A| t = |∂A|.

Connectedness properties of the parallel set

Let A be a bounded measurable subset of R n and B be a convex body in R n , then for every t > 0 the set A + tB has a finite number of connected components and this number is non-increasing as a function of t. Indeed, let t > 0 and C be a connected component of

A + tB. Let x ∈ C, then there exists a ∈ A such that x ∈ a + tB. Moreover a + tB is connected, hence a + tB ⊂ C since C is the connected component of x. Thus |C| ≥ |tB| > 0.
Since |A + tB| is finite and equal to the volume of the disjoint union of its connected components, there is a finite number of them.

Let

0 < t 0 ≤ t 1 . Denote by C 1 , . . . , C N the connected components of A + t 0 B. One has A + t 1 B = ∪ N i=1 (C i + (t 1 -t 0 )B) and since C i + (t 1 -t 0 )B is connected, it follows that the number of connected components of A + t 1 B is at most N .

Regularity properties of the parallel volume

Let A be a compact subset of R n and B be a convex body in R n containing 0 in its interior. The function d B (•, A) is Lipschitz, hence from Federer's co-area formula [START_REF] Federer | Geometric Measure Theory[END_REF], one has

V A,B (t) = |A + tB| = |A| + t 0 H n-1 ({x : d B (x, A) = s})ds, (1)
where H n-1 denotes the (n -1)-dimensional Hausdorff measure. Therefore the function V A,B is absolutely continuous on R + . Notice that for every bounded measurable subset A of R n and every 0 < s < t, one has

A + sB ⊂ A + tB ⊂ A + tB.
From the continuity of V A,B , one gets that |A + tB| = |A + tB| for t > 0.

Hence we may assume in the following that A is compact.

Stachó [START_REF] Stachó | On the volume function of parallel sets[END_REF] proved a better regularity for V A,B , he proved namely that the function V A,B is a n-Kneser function, which means that for every 0 < t 0 ≤ t 1 and every λ ≥ 1, one has

V A,B (λt 1 ) -V A,B (λt 0 ) ≤ λ n (V A,B (t 1 ) -V A,B (t 0 )). (2)
Stachó deduced that for every 0 < t 0 < t 1 , the function +∞). Thus V A,B admits right and left derivatives at every t > 0, which satisfy

t → V A,B (t) -t n V A,B (t 1 ) -V A,B (t 0 ) t n 1 -t n 0 is concave on [t 1 ,
(V A,B ) ′ + (t) ≤ (V A,B ) ′ -(t) (3)
and these two derivatives coincide for all t > 0 outside a countable set. Hence the outer Minkowski surface area of A + tB n 2 exists for every t > 0 and one has

|∂(A + tB n 2 )| = lim ε→0 + |A + tB n 2 + εB n 2 | -|A + tB n 2 | ε = (V A ) ′ + (t). (4)
In Proposition 3.8 below, we show that the function

V A is continuously dif- ferentiable on [diam(A), +∞). If A is
|A + tB| ≤ |A + t ′ B| ≤ |A + tB + r(t ′ -t)B n 2 |.
From (1) the function t ′ → |A + tB + t ′ rB n 2 | is continuous at 0, thus the function t → |A + tB| is continuous on R + . Since for s > 0 and t ≥ 0

|sA + tB| = s n A + t s B
then (s, t) → |sA + tB| is continuous on (0, +∞) × R + . We also have for any s ≥ 0 and t ≥ 0

|tB| ≤ |sA + tB| ≤ |srB n 2 + tB| so (s, t) → |sA + tB| is continuous on {0} × R + . It follows that the function (s, t) → |sA + tB| is continuous on R + × R + .
The monotonicity follows from [START_REF] Ambrosio | Outer Minkowski content for some classes of closed set[END_REF]. Indeed, the inequality (2) may be written in a different way, as follows

|A + λt 1 B| -|A + λt 0 B| ≤ |λA + λt 1 B| -|λA + λt 0 B|.
Changing variables, it also means that for every 0 < s 0 ≤ s 1 and 0 < t 0 ≤ t 1

|s 0 A + t 1 B| -|s 0 A + t 0 B| ≤ |s 1 A + t 1 B| -|s 1 A + t 0 B|.
Applied first to s 1 = 1 and s 0 → 0, and then to t 1 = 1 and t 0 → 0, we deduce that the functions t → V A,B (t) -t n |B| and s → |sA + B| -s n |A| are non-decreasing. In particular, the function (s, t) → |sA + tB| is nondecreasing in each coordinate.

Remark. If A and B are arbitrary compact sets, it is not necessarily true that the function V A,B is non-decreasing as can be seen from the example of A = {0; 4} and B = [-5, -3] ∪ [START_REF] Blachman | The convolution inequality for entropy powers[END_REF][START_REF] Costa | A new entropy power inequality[END_REF].

Links with geometric inequalities

Let us connect the Costa-Cover conjecture with the Brunn-Minkowski inequality and the isoperimetric inequality. We first establish that the conjecture of Costa-Cover has many equivalent reformulations. Proposition 2.2. Let A and B be compact sets in R n , with B convex. The following properties are equivalent.

(i) t → |A + tB| 1 n is concave on R + . (ii) s → |sA + B| 1 n is concave on R + . (iii) λ → |(1 -λ)A + λB| 1 n is concave on [0, 1]. (iv) (s, t) → |sA + tB| 1 n is concave on R + × R + .
Proof. (iv)=⇒(i), (iv)=⇒(ii) and (iv)=⇒(iii) are clear. Let us prove that (i)=⇒(iv), a similar argument easily shows that (ii)=⇒(iv) and (iii)=⇒(iv). Let f : R + → R and g : R + × R + → R be defined by f (t) = |A + tB| 1 n and g(s, t) = |sA + tB| 1 n , for every s, t ∈ R + . For every t ≥ 0 and s > 0, we have, from the homogeneity of the volume g(s, t) = sf t s .

Thus for every λ ∈ [0, 1], s 1 , s 2 ∈ (0, +∞) and t 1 , t 2 ∈ R + we get

g((1 -λ)s 1 + λs 2 , (1 -λ)t 1 + λt 2 )) = ((1 -λ)s 1 + λs 2 )f (1 -λ)t 1 + λt 2 (1 -λ)s 1 + λs 2 .
Using the concavity of f , we deduce that

f (1 -λ)t 1 + λt 2 (1 -λ)s 1 + λs 2 = f (1 -λ)s 1 t 1 s 1 + λs 2 t 2 s 2 (1 -λ)s 1 + λs 2 ≥ (1 -λ)s 1 f t 1 s 1 + λs 2 f t 2 s 2 (1 -λ)s 1 + λs 2 = (1 -λ)g(s 1 , t 1 ) + λg(s 2 , t 2 ) (1 -λ)s 1 + λs 2 .
We deduce that g is concave on (0, +∞) 2 . Moreover, g is continuous on (R + ) 2 by Proposition 2.1. Hence g is concave on (R + ) 2 .

Remark. Notice that if for two fixed compact sets A and B, with B convex, the assertion (iii) of Proposition 2.2 holds true then for every λ ∈ [0, 1],

|(1 -λ)A + λB| 1 n ≥ (1 -λ)|A| 1 n + λ|B| 1 n
, which is the Brunn-Minkowski inequality. Hence the conjecture of Costa-Cover ((i) of Proposition 2.2) implies the Brunn-Minkowski inequality in the case where one set is convex.

Let us study the connection with the isoperimetric inequality. The Costa-Cover conjecture implies that for every t ≥ 0 and every sufficiently regular compact set A

1 n |∂A| n-1 |A| 1-1 n = (V 1/n A ) ′ + (0) ≥ (V 1/n A ) ′ + (t) ≥ lim t→+∞ (V 1/n A ) ′ + (t) = 1 n |∂B n 2 | n-1 |B n 2 | 1-1 n ,
which is the isoperimetric inequality. This would give a non-increasing path from

|∂A| n-1 |A| 1-1 n to |∂B n 2 | n-1 |B n 2 | 1-1 n through the family |∂(A + tB n 2 )| n-1 |A + tB n 2 | 1-1 n t∈R + .
We may apply the same arguments for any convex body B instead of B n 2 . Thus, the conjecture that t → V A,B (t) 1/n is concave on R + implies the following generalized isoperimetric inequality, also known as Minkowski's first inequality proved for example in [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory, encyclopedia of mathematics and its applications 44[END_REF],

|∂ B A| n-1 |A| 1-1 n ≥ |∂ B B| n-1 |B| 1-1 n = n|B| 1 n .
3 The 1 n -concavity of the parallel volume

Recall that for t ≥ 0, V A (t) = |A+tB n 2 | and that Costa and Cover [START_REF] Costa | On the similarity of the entropy power inequality and the Brunn-Minkowski inequality[END_REF] conjectured the 1 n -concavity of V A on R + , for every compact A. They also noticed that their conjecture holds true for A being convex. Let us repeat their argument. For every λ ∈ [0, 1] and t, s ∈ R + , from the Brunn-Minkowski inequality, one obtains

|A + ((1 -λ)t + λs)B n 2 | 1 n = |(1 -λ)(A + tB n 2 ) + λ(A + sB n 2 )| 1 n ≥ (1 -λ)|A + tB n 2 | 1 n + λ|A + sB n 2 | 1 n .
Notice that from the same argument we deduce that for every convex sets A and B, the function V A,B (t) = |A+tB| is 1 n -concave on R + . Hence for convex sets A and B, the properties (i)-(iv) of Proposition 2.2 holds true. In this case, the 1 n -concavity of V A,B on R + is equivalent to the Brunn-Minkowski inequality (and true).

In dimension 1

Let us prove the Costa-Cover conjecture in dimension 1.

Proposition 3.1. Let A be a compact set in R and B be a convex body in R, then t → V A,B (t) = |A + tB| is concave on R + .

Proof. We note that in dimension 1, for t 0 > 0, A + t 0 B is a disjoint finite union of intervals. Thus, by setting A + t 0 B for an arbitrary t 0 > 0 instead of A, we can assume that

A = ∪ N i=1 [a i , b i ], with a i , b i ∈ R, N ∈ N * . Thus, for t sufficiently small, V A,B (t) = |A + tB| = N i=1 (b i -a i + |B|t) = N i=1 (b i -a i ) + |B|N t.
Thus V A,B is piecewise affine on R * + . Moreover, when t increases, the slope of V A,B is non-increasing since the number of intervals composing A + tB is non-increasing. Using that V A,B is continuous on R + , we conclude that it is concave on R + .

Remarks. For arbitrary compact sets A and B, function V A,B is not necessarily concave as can be seen from the example of A = {0; 4} and B = [-5, -3] ∪ [START_REF] Blachman | The convolution inequality for entropy powers[END_REF][START_REF] Costa | A new entropy power inequality[END_REF], the same example which was given in the remark after Proposition 2.1 to show that the function V A,B is not necessarily increasing.

In dimension 2

We first prove the Costa-Cover conjecture for compact connected sets in dimension 2.

Theorem 3.2. Let A be a compact subset of R 2 . Then, V A : t → |A + tB 2 2 | is 1
2 -concave on R + . Proof. We proceed by approximating A by finite sets, hence let us first assume that A is finite,

A = {x 1 , . . . , x N }. Let T = {t 1 , . . . , t m } ⊂ R + , with t 1 < • • • < t m ,
be the finite set of real numbers which are equal to

|x i -x j | 2
for some i, j ∈ {1, . . . , N } or to the radius of the circumscribed circle of a triangle (x i , x j , x k ) for some i, j, k ∈ {1, . . . , N }. For t > 0, let p A (t) be the number of connected components of A + tB 2 2 and q A (t) be the genus of A + tB 2 2 . Notice that the functions p A and q A are piecewise constants on R + \ T and that V A is infinitely differentiable on R + \ T , (see proposition 4.8 in [START_REF] Gorbovickis | Strict Kneser-Poulsen conjecture for large radii[END_REF]).

We use a key result established by Fiala in the context of Riemannian manifolds, see [START_REF] Fiala | Le problème des isopérimètres sur les surfaces ouvertes à courbure positive[END_REF], first part, section 9 "vraies parallèles": for every t ∈ (0, +∞) \ {t 1 , . . . , t m },

V ′′ A (t) ≤ 2π(p A (t) -q A (t)).
Notice that p A (t) -q A (t) is equal to the Euler-Poincaré characteristic of A + tB 2 2 . Now, we consider t 0 ∈ R + such that A+t 0 B 2 2 is connected. Then for every t ≥ t 0 , A + tB 2 2 is connected. Hence for every t ∈ (t 0 , +∞) \ {t 1 , . . . , t m },

V ′′ A (t) ≤ 2π. (5)
Let us prove that V A is 1 2 -concave on (t 0 , +∞). By the isoperimetric inequality, we have for every t ∈ (t 0 , +∞) \ {t 1 , . . . , t m },

4π|A + tB 2 2 | ≤ |∂(A + tB 2 2 )| 2 ,
we write this in this form

4πV A (t) ≤ V ′ A (t) 2 ,
thus, using (5),

2V A (t)V ′′ A (t) ≤ V ′ A (t) 2 . Hence √ V A ′′ (t) ≤ 0. We conclude that V A is 1 2 -concave on (t i , t i+1
), for all i ≤ m -1 and on (t m , +∞). From (3) we have

(V A ) ′ -(t i ) ≥ (V A ) ′ + (t i ), thus V A is 1
2 -concave on (t 0 , +∞). Let us then consider a compact connected set A of R 2 . Let t 0 > 0. Let (x N ) N ∈N * be a dense sequence in A. We denote, for N ∈ N * , A N = {x 1 , . . . , x N }. There exists N 0 ∈ N * such that for every N ≥ N 0 , A N + t 0 B 2 2 is connected. For every N ≥ N 0 , we have shown that V A N is 1 2 -concave on (t 0 , +∞). Moreover the sequence (A N ) N → A in the Hausdorff distance, thus by denoting d N = d H (A N , A), the Hausdorff distance, one has, for every t > 0

A N + tB 2 2 ⊂ A + tB 2 2 ⊂ A N + (t + d N )B 2 2 .
Applying the right hand side inclusion to t replaced by t -d N where N satisfies d N < t, we deduce

A + (t -d N )B 2 2 ⊂ A N + tB 2 2 ⊂ A + tB 2 2 .
Hence by continuity of the function V A at the point t,

lim N →+∞ V A N (t) = V A (t).
It follows that √ V A is the pointwise limit of a sequence of concave functions, hence V A is 1 2 -concave on (t 0 , +∞), for every t 0 > 0. We conclude that V A is 1 2 -concave on R + .

Proposition 3.6. Let A be a compact subset of R n . Then the function t → |A + tconv(A)|

u = 2x -y ∈ [y, z] ⊂ conv(A). Hence x = y + u 2 ∈ ∂conv(A) + conv(A) 2 . Finally conv(A) ⊂ ∂conv(A) + conv(A) 2 ⊂ A + conv(A) 2 ⊂ conv(A).
We deduce that A+tconv(A) = (1+t)conv(A), for every t ≥ 1. We conclude that

t → |A + tconv(A)| 1 n is affine on [1, +∞).
Remark. More generally, Schneider introduced in [START_REF] Schneider | A measure of convexity for compact sets[END_REF] the quantity

c(A) = inf{t ≥ 0 : A + tconv(A) = (1 + t)conv(A)}. Clearly t → |A + tconv(A)| 1
n is affine on [c(A), +∞). The above proposition establishes that c(A) ≤ n in general and c

(A) ≤ 1 if ∂conv(A) ⊂ A. Notice that if A ⊂ R n is connected then c(A) ≤ n -1, see [19]. Theorem 3.7. Let A be a compact set in R n . If the function ε → |εA + B n
2 | is twice differentiable in a right neighborhood of 0, with second derivative continuous at 0 + , then there exists t 0 ≥ 0 such that the function V A (t) = |A + tB n 2 | is 1 n -concave for t ≥ t 0 . In particular this holds for A being finite.

Proof. Kampf proved in [START_REF] Kampf | Asymptotic order of the parallel volume difference[END_REF], lemma 28, that for every compact set A there exists a constant C which depends on n, A so that for every t ≥ 1,

0 ≤ |conv(A) + tB n 2 | -|A + tB n 2 | ≤ Ct n-3 .
Then, setting ε = 1 t , for every ε ∈ (0, 1], one deduces

0 ≤ |εconv(A) + B n 2 | -|εA + B n 2 | ≤ Cε 3 . (6)
We denote g

conv(A) (ε) = |εconv(A) + B n 2 | and g A (ε) = |εA + B n 2 |, since g A is twice differentiable at 0 + it follows that g A (0) = g conv(A) (0) , (g A ) ′ + (0) = (g conv(A) ) ′ + (0) , (g A ) ′′ + (0) = (g conv(A) ) ′′ + (0).
1. For an arbitrary compact subset A of R 3 , if there exists a sequence (x N ) N ∈N * dense in A such that for every N , the set A N satisfies the condition (⋆), where A N = {x 1 , . . . , x N }, and such that t 0 (A N ) is uniformly bounded in N by a t 0 , then the function t → |A + tB 3 2 | will be 1 3 -concave on [t 0 , +∞). 2. In dimension n ≥ 4, there is no hope to prove the inequalities [START_REF] Cover | Information theoretic inequalities[END_REF] because for A being two points at distance 2, one has for every t ≥ 1

V ′ A (t) = n|B n 2 |t n-1 + 2(n -1)|B n-1 2 |t 1 0 (t 2 -x 2 ) n-3 2 dx < n|B n 2 |t n-1 + 2(n -1)|B n-1 2 |t n-2 = V ′ conv(A) (t).

Further analogies

In Information theory, the Blachman-Stam inequality ( [START_REF] Blachman | The convolution inequality for entropy powers[END_REF] and [START_REF] Stam | Some inequalities satisfied by the quantities of information of Fisher and Shannon[END_REF]), which states that for any independent random vectors X and Y in R n with non-zero Fisher information one has

I(X + Y ) -1 ≥ I(X) -1 + I(Y ) -1 ,
directly implies all previous mentioned inequalities of Information theory: the entropy power inequality (thus the Log-Sobolev inequality for Gaussian measure) and the concavity of entropy power. This last inequality also called the "isoperimetric information inequality" may be deduced from the Blachman-Stam inequality in the same way as the "isoperimetric entropy inequality" was deduced from the entropy power inequality, by applying it to Y = √ εG and letting ε tend to 0.

Let us now investigate the analogue of the Fisher information and the Blachman-Stam inequality in the Brunn-Minkowski theory. Recall de Bruijn's identity

I(X) = d dt |t=0 2H(X + √ tG).
Since the entropy H is the analogue of the logarithm of the volume log | • |, Dembo, Cover and Thomas [START_REF] Cover | Information theoretic inequalities[END_REF] proposed, as an analogue of the Fisher information I, the quantity

d

  convex or with sufficiently regular boundary then the equality (4) also holds for t = 0. For precise statements and comparisons between the outer Minkowski surface area and other measurements of ∂(A + tB n 2 ), like the Hausdorff measure, see[START_REF] Ambrosio | Outer Minkowski content for some classes of closed set[END_REF].

Proposition 2.1. Let A and B be compact subsets of R n with B convex, then the function (s, t) → |sA + tB| is continuous on R + × R + . Moreover the functions t → |A + tB| -t n |B| and s → |sA + B| -s n |A| are non-decreasing. In particular, the function (s, t) → |sA + tB| is nondecreasing in each coordinate. Proof. Let us prove the continuity. Let 0 ≤ t ≤ t ′ . Let r > 0 be such that A ⊂ rB n 2 and B ⊂ rB n 2 . Then we have

  1/n is affine on [n, +∞). If moreover ∂conv(A) ⊂ A then this function is affine on [1, +∞). If moreover ∂conv(A) ⊂ A then for every x ∈ conv(A) there exists two points y, z in ∂conv(A) such that x ∈ [y, z]. Say, for example, that |x -y| ≤ |x -z| then

	Proof. It was noticed by Schneider [19] that A + tconv(A) = (1 + t)conv(A),
	for every t ≥ n. Thus t → |A + tconv(A)|	1 n is affine on [n, +∞).

  dε |ε=0 (log |A + εB n 2 |) = |∂A| |A| ,for sufficiently regular compact sets A. Thus, in analogy with the Blachman-Stam inequality, one may wonder if for every regular compact sets A and B

	|A + B| |∂(A + B)|	≥	|A| |∂A|	+	|B| |∂B|	.	(8)

Acknowledgement.

We thank Evgueni Abakoumov, Ludovic Goudenège and Olivier Guédon for their interesting remarks.

The authors were supported in part by the Agence Nationale de la Recherche, project GeMeCoD (ANR 2011 BS01 007 01).

Remarks

1. In the proof of Theorem 3.2, from the bound V ′′ A (t) ≤ 2π(p A (t)-q A (t)) obtained for every finite set A and for every t > 0 outside a finite number of points, one deduces that for every compact subset A of R 2 with finite connected components p A , the function t → V A (t) -p A πt 2 is concave on (0, +∞). From Steiner's formula one has

If A is connected, it follows that

is convex as the sum of an affine function and a convex function. Notice that this complements the result of Kampf [START_REF] Kampf | The parallel volume at large distances[END_REF] who proved that V conv(A) (t) -V A (t) tends to 0 as t → +∞.

2. If in Theorem 3.2 we replace B 2 2 by an ellipsoid, i.e. by T (B 2 2 ) where T is an invertible linear transformation, then the result holds since

For a non-connected set A, the next proposition shows that the function

2 ), we have

Since the 1 n -power of this function is not concave (it is strictly convex),

Remark. This counterexample shows that the Brunn-Minkowski inequality doesn't imply the 1 n -concavity of the parallel volume for non convex sets.

In dimension n ≥ 3

We may ask if the Costa-Cover conjecture still holds for connected sets in dimension n ≥ 3. The next proposition shows that this is false: even for star-shaped body, the function

x 1 ≥ 0}. We use Steiner's formula for each term. One has

For the second term, we first notice that

Using for example [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory, encyclopedia of mathematics and its applications 44[END_REF] p. 294 formula (5.3.23), we get that the coefficient of t 2 in the Steiner expansion of

. The third term is equal to t 3 times a polynomial. Thus, there are coefficients a 0 , . . . , a n such that for t ∈ [0, 1],

is not concave in a neighborhood of 0.

We have seen that the Costa-Cover conjecture does not hold in general. We still conjecture that the following weaker form may hold. Conjecture 3.5. Let A be a compact subset of R n and B be a convex body in R n . Then there exists t 0 such that the function

We have shown that this conjecture is true in dimension 1 and in dimension 2 for B = B 2 2 . Indeed, in dimension 2, we have seen that it is true for every compact connected set. Since for every compact subset

We prove the Conjecture 3.5 in some particular cases in dimension n ≥ 3.

From Steiner's formula, we get g conv(A) (0) = |B n 2 | and

If conv(A) is not homothetic to B n 2 , then from the equality case of the Alexandrov-Fenchel inequality, see [START_REF] Schneider | Convex bodies: the Brunn-Minkowski theory, encyclopedia of mathematics and its applications 44[END_REF], theorem 6.6.8, page 359, we get

Thus we deduce that

Since g A , g ′ A and g ′′ A are continuous at 0 + , there exists

Hence the function g

2 then the result follows from Proposition 3.6. If A is finite then the function ε → |εA + B n 2 | is analytic in a right neighborhood of 0, see [START_REF] Gorbovickis | Strict Kneser-Poulsen conjecture for large radii[END_REF].

Remarks.

The preceding theorem is still valid if one replaces B n

2 by a convex body B = rB n 2 + M , for some r > 0 and some convex body M such that its support function h B (u) = max{< x, u >, x ∈ B} is twice differentiable on R n \ {0} because, as proved in [START_REF] Kampf | Asymptotic order of the parallel volume difference[END_REF], inequality [START_REF] Costa | On the similarity of the entropy power inequality and the Brunn-Minkowski inequality[END_REF] holds with these assumptions.

The function

is not necessarily twice differentiable in a neighborhood of 0 as can be seen from the following example. In dimension 2, we consider the points I = (1, 1), J = (1, 0) and

| is not twice differentiable on (t 0 , +∞).

In fact, one can show that the function V

Proof. Rataj et al. in [START_REF] Rataj | On the expected surface area of the Wiener sausage[END_REF], theorem 3.3, showed that V ′ A (t) exists for every t ≥ diam(A), thus we have for every t ≥ diam(A)

Moreover, if (A N ) is a sequence of non-empty compact subset of R n tending in Hausdorff distance to a compact subset A of R n , then by [START_REF] Stachó | On the volume function of parallel sets[END_REF], theorem 3, for every t > 0 such that

Let t ≥ diam(A), we apply this result to

, where (t N ) is a sequence of non-negative numbers tending to 0. We obtain that lim

Hence, V ′ A is right continuous at t. Let t, t 0 be such that t > t 0 > diam(A), we now apply the result of Stachó to

2 , where (t N ) is a sequence of non-negative numbers tending to 0. We obtain

Moreover, from the inequality (6), valid for any compact set A, one deduces that g A is also differentiable at 0, with g

A special case in dimension 3

We have seen that for every finite subset A of R n , there exists t 0 (A) such that the function

In dimension 3, we can give a bound on t 0 (A) in terms of the geometry of A.

In the sequel, A denotes a finite subset of R 3 . We denote by D i a Dirichlet-Voronoi cell with respect to A = {x 1 , . . . , x N }, defined for i ∈ {1, • • • , N } by

The following condition can be found in [START_REF] Kampf | Large parallel volumes of finite and compact sets in d-dimensional Euclidean space[END_REF].

Condition (⋆) For all faces F of the polytope conv(A), and all edges E of F , we have

For example, if conv(A) is simplicial, this condition holds if and only if each face of conv(A) is a triangle with only acute angles. In general, this condition holds if and only if for every face F of conv(A), for every edge [a, b] of F and for every vertex c of F , the angle (ca, cb) is acute. Proposition 3.9. Let A be a finite set in R 3 satisfying the condition (⋆).

Proof. Kampf and Kiderlen have shown in [START_REF] Kampf | Large parallel volumes of finite and compact sets in d-dimensional Euclidean space[END_REF] that for every t > t 0 (A),

with for all p ≥ 0, a p ≥ 0. Since V conv(A) is polynomial thus V A is twice differentiable on (t 0 (A), +∞). It follows that for every t > t 0 (A),

Then, for every t > t 0 (A),

The Brunn-Minkowski inequality implies that V conv(A) is 1 3 -concave on R + . We conclude that for every t > t 0 (A),

So, V A is 1 3 -concave on [t 0 (A), +∞).

Remarks.

Even restricted to the case where A and B are convex sets, checking the validity of this inequality is not an easy task and it was conjectured by Dembo, Cover and Thomas [START_REF] Cover | Information theoretic inequalities[END_REF] that the inequality (8) holds true in this particular case. In [START_REF] Fradelizi | Some inequalities about mixed volumes[END_REF], it was shown that this conjecture (for convex sets) holds true in dimension 2 but is false in dimension n ≥ 3. In particular, it was proved that, if n ≥ 3, there exists a convex body K such that the inequality (8) cannot be true for all A, B ∈ {K + tB n 2 : t ≥ 0}. It was also proved that if B is a segment then there exists a convex body A for which (8) is false.

In another direction, one may also ask if [START_REF] Dembo | Information inequalities and uncertainty principles[END_REF] holds true for B being any Euclidean ball and every compact set A. In this case, applying [START_REF] Dembo | Information inequalities and uncertainty principles[END_REF] with the notations given above, this would mean that

is non-decreasing on (0, +∞). This is equivalent to the 1 n -concavity of V A , which is the Costa-Cover conjecture.

Extensions.

The second named author pursued these investigations in [START_REF] Marsiglietti | Concavity properties of extensions of the parallel[END_REF]. More precisely, he discusses the concavity properties of the function t → µ(A + tB n 2 ), where µ is a log-concave measure. He also establishes functional versions of Costa-Cover conjecture.