Thomas Streubel
email: thomas.streubel@gmail.com

Pierre De Beaucorps
email: pierre.de-beaucorps@inria.fr

Fawzi Nashashibi
email: fawzi.nashashibi@inria.fr

Evaluation of automated vehicle behavior in intersection scenarios

Keywords: Multi-agent simulation, subjective risk assessment, automated vehicle behavior, automated vehicle evaluation

The development of automated vehicles (AVs) is ongoing and soon the first versions will enter the up to now exclusively human controlled traffic environment. In previous research, we developed an approach for an automated vehicle control in intersection scenarios. The focus was on the decision-making process to either enter the intersection before another car or wait to let the other car pass. While objective risk features were used to evaluate the performance, the interaction with real human drivers remains uncertain. Therefore, we propose a method to evaluate the automated vehicle behavior from an outside perspective by introducing a human driver and our automated vehicle in the same simulation environment. We conducted a study to examine the interaction and evaluate the automated vehicle behavior from another driver's perspective. The focus of the study was on the naturalness and risk of the behavior and how the subjective risk evaluation is linked to our main objective risk feature. For comparing the results, we used two different setups for our automated vehicle (aggressive vs. passive) and varied the starting position to create either of the two situations (AV first vs. driver first) as mentioned above. The results show a high overall naturalness score. The risk score was related to the outcome of the situation and the AV setup. A weak correlation between the subjective risk assessment and the objective risk feature indicated the necessity to evaluate automated vehicle control from an outside perspective to achieve a better assimilation in human traffic environments. Since the identification of automated vehicles can influence the driver behavior, the integration of a human-like behavior for AVs seems highly desirable.

Introduction

On the verge of automated vehicles (AVs) entering the existing traffic environment new challenges arise. This concerns especially the interaction with human driven vehicles which is subject of present research. Here, the focus is mainly on the reaction of the automated vehicle towards other human controlled cars in real traffic environments and particularly the performance within certain comfort limits to satisfy passenger's expectations inside the AV. The performance from an outside perspective especially its ability to "blend in" is less investigated. However, the development of automated vehicle control requires intense testing and validation when it comes to interaction with other vehicles. This is commonly conducted against predefined safety and comfort metrics but also requires subjective testing when integrated into a human domain such as the traffic environment. The latter point is particularly relevant when the presence of an automated vehicle is influencing the human drivers' behavior [START_REF] Aria | Investigation of Automated Vehicle Effects on Driver's Behavior and Traffic Performance[END_REF]. In longitudinal traffic, this influence was shown in [START_REF] Gouy | Driving next to automated vehicle platoons: How do short time headways influence non-platoon drivers' longitudinal control?[END_REF]. A low time headway (THW) in a platoon of automated vehicles led to a reduction of the THW by the driver even below a safety threshold of 1.0s. This behavior adaptation is safety critical and requires more awareness especially since modelling driving behavior is only based on the current state of human controlled traffic environments. We propose a method to evaluate the automated vehicle behavior in intersection scenarios. Intersections are challenging elements of the traffic network with a high level of interaction between traffic participants leading to high numbers of accidents. Due to this danger, intersection scenarios with AVs are best investigated in simulation. Therefore, a simulation environment was created where human drivers can interact with others and also with automated vehicle. In previous research, we developed a setup for an automated control focusing on the decision making process at intersections [START_REF] Beaucorps | Decisionmaking for automated vehicles at intersections adapting human-like behavior[END_REF]. The situation when two vehicles approach an intersection can dissolve into two possible outcomes (besides a collision). We use the post-encroachment time (PET) as a risk feature for the decision process. This time refers to a conflict zone that both approaching vehicles will pass when crossing the intersection. The PET is mentioned in [START_REF] Bärgman | Quantifying drivers' comfort-zone and dread-zone boundaries in left turn across path/opposite direction (LTAP/OD) scenarios[END_REF] as a parameter related to risk for left turning scenarios. It is defined as the time between one vehicle leaving the conflict zone and the other one entering this zone. The study examined the driving behavior in a real driving scenario under two conditions of the driver (comfort vs. hurried) to explore comfort limits of the PET. The results showed mean values between 1.5s (hurried condition) and 2.2s.

In this paper, we investigate the interaction between one human driver and an automated vehicle at an X-shaped intersection. We conducted a simulator study to evaluate the performance and behavior of our automated vehicle subjectively and compare this with our objective risk evaluation. The naturalness and the risk assessment of the human drivers towards the AV were focused on. As objective risk feature, we use the PET.

Methodology

Simulation environment

The simulation environment used in this study is self-designed in JavaScript as a multi-agent browser application with client-server architecture. This way, several drivers can interact in the same simulation environment as well as with our automated vehicle. The simulator tool is easily accessible through a browser without any installation. A 3D view and a bird-view are shown in Figure 1. In the drivers perspective the assigned path is highlighted. To reduce the complexity of the situation, the human controlled car was always going straight at the intersection. The simulation is based on the open source bullet physics engine [5]. The vehicle model is an adaptation of the Raycast vehicle [START_REF] Maddock | Vehicle Simulation with Bullet[END_REF] with a suspension system for each wheel and a body mass as chassis above. This realizes pitch and bounce in the simulation giving a realistic impression of rapid changes in speed to the human driver. The computation of the physical motion of the vehicles is centralized on a server while the client hardware is only responsible for the visual representation. This saves computation resources and enables the usage of the simulation with multiple clients. For the control of the vehicle, there are a steering wheel and pedals for realistic inputs. However, while the steering wheel was set up in front of the participants to create a car like atmosphere, it remained unused because of the predefined straight path. The lateral control was automated since the usage of steering within the simulation requires some training and fine-tuning. The pedals were used to control the acceleration and brake. Shifting gears was unnecessary since the vehicles are considered automatic. For the visual output, we used three monitors arranged slightly concave. This realizes a viewing angle of about 114 degrees. Rearview mirrors are not implemented.

Participants

Overall 16 drivers participated in the simulator study. Three of them were female. The average age was 30.4 years (sd= 9.1 years). Everyone was in possession of a valid driver's license in average for 10.1 years (sd= 7.3 years). The mean annual travel distance was around 6,000 km with about half of it in urban environments.

Scenarios

We created a standard X-shaped 4-way crossroad with one lane entering and exiting in each direction. The test drivers were passing the intersection going straight. Hence, there are six scenarios how an automated vehicle could interact with the drivers by interfering with their path (cf. Figure 2). The conflict can arise either by crossing paths or by merging into the same direction which can result in different incidents. The scenarios are distinguished by the maneuver of the AV combined with the potential conflict: left turn across path (LTAP), left turn same paths (LTSP), right turn same paths (RTSP) and straight across path (SAP). The approaching direction relative to the human driver is the second criteria to describe a scenario: left direction (LD), right direction (RD) and opposite direction (OD). All combinations with interaction result in the six scenarios: two for left turn across path (LTAP-RD and LTAP-OD), one left turn same paths (LTSP-LD), two for straight across path (SAP-RD and SAP-LD) and one right turn same paths (RTSP-RD) scenario (see Figure 2). The traffic signs were placed so the drivers had always the priority. Hence, the automated vehicle should yield unless passing before the driver is safe and is not interfering with the trajectory of the human driver. These situations were especially of interest in the study since a car entering in front of the human can be experienced as unnatural or risky especially when it forces a reaction from the driver. To create those situations with interaction the starting position of the AV was varied.

Automated Vehicle Control

A detailed explanation of the automated vehicle control and decision algorithm is given in [START_REF] Beaucorps | Decisionmaking for automated vehicles at intersections adapting human-like behavior[END_REF]. Here, we give a brief overview. The decision algorithm includes a simple prediction of the other vehicle. Further, we extracted several speed profiles from real drivers driving in our simulation. These speed profiles are clustered and serve as proposals for the decision algorithm. The predicted trajectory of the AV is calculated by utilizing these reference speed profiles as target speed taking into account the actual position relative to the intersection and the present velocity. Here, the path is preset since the scenarios are predefined in the study design (see Scenarios). The retrieved trajectory options are now compared to the extrapolation of the other vehicle (constant speed assumption). The prediction horizon is set to about 3 seconds. When both vehicles approach the intersection, the predicted future vehicle positions intersect and a PET value can be estimated. This is the main risk feature in the algorithm. For the scenarios where both vehicles drive in the same direction after passing the intersection, the definition of the conflict zone is extended since the paths are merging. Here, the conflict zone to determine the PET is starting when both vehicle positions meet for the first time but then is limited to a car length. Without this restriction, the conflict zone would extend over the entire further lane since both cars drive on the same leg of the intersection. A positive PET is referring to the AV passing before the other vehicle. If the estimated PET value drops below a certain threshold, the associated trajectory is discarded. The decision algorithm is eventually choosing the trajectory with an estimated PET value higher than the threshold and with the highest overall velocity to optimize the travel time. Every time step (about 100ms) all trajectories are tested this way. For cases where all speed profiles are risky, there is a stopping trajectory as fall back option. As soon as the vehicle enters the conflict zone, it passes the intersection quickly and switches into a car following (ACC) mode. This prevents dangerous situations by stopping in the intersection area or following the driver in a too close distance.

Procedure

At the beginning of the study the participants received the instructions and filled out a short demographic survey. Afterwards, they were able to get used to the vehicle control and the simulation environment in a small test track. When the driver felt comfortable with the pedals, the study started. The intersections were presented to the driver consecutively in a randomized order to avoid sequence effects. As mentioned before, the interaction between AV and driver was desirable. However, since the approaching behavior of the drivers is dependent on individual preferences, a synchronization to standardize the scenarios is nearly impossible. The AV was early visible in our open field design, so late manipulation of the AV could affect the assessment of the driver and also would be in contradiction to the algorithm for the automated vehicle control that aims to avoid high risk situations. However, to avoid delays at the start, the AV is triggered by the first input of the driver (usually pushing the gas pedal). To realize different situations with AV passing before the human driver and also yielding, we chose three different starting positions for the AV fine-tuned for every scenario. Finally, the simulation stops shortly after the driver passed the intersection. Since the path was straight, the driver could focus on the longitudinal control with the driving pedals. The path was also visible in the scene for the driver to assure the driving direction (see Figure 1 right side). After each scenario, the drivers were asked about the naturalness of the behavior of the other vehicle, how risky they rank this behavior and if they would change their own behavior retrospectively.

Design

The study was conducted in a repeated-measurement design. All six scenarios were presented to every driver in a randomized order. There were two different automated vehicle setups (aggressive vs. passive) to vary the behavior of the AV. This was realized by changing the PET threshold in the algorithm. The aggressive setup accepts PET values down to 0.5 seconds while the passive one only passes before the human driver if the estimated PET is above 1.5 seconds. This value is chosen in compliance with the results in [START_REF] Bärgman | Quantifying drivers' comfort-zone and dread-zone boundaries in left turn across path/opposite direction (LTAP/OD) scenarios[END_REF] for the hurried human driver. Previous tests have shown that even higher thresholds lead to a very conservative behavior and almost always result in a yield to oncoming vehicles. The starting distance of the AV was varied between three points to create interaction in the scenarios. Further, the vehicle dynamics were recorded for every run enabling a replay afterwards to confirm certain actions. For the evaluation, the actual PET was determined for every case. Here, a positive PET indicates that the AV was passing before the driver and negative PET values imply the opposite. Accordingly, low absolute PET values can be seen as more dangerous than higher absolute values. In the short questionnaire after every setup, the naturalness and riskiness of the other vehicle was ranked by the test drivers.

Analysis and Results

The driving data along with the answers to the questionnaire were stored during the study by the simulation tool into json-files. For the analysis, this data is converted into Matlab. Overall, the test drivers crossed the intersection 576 times (16 drivers * 2 AV setups * 6 scenarios * 3 variations). The three variations refer to different starting points for the AV to create interaction in the scenarios. In the process, the AV passed 235 times first (40.8%). This situation occurred more often in the aggressive AV setup as shown in Table 1. There were two incidents with the aggressive AV in which the PET values were zero; one was an actual collision where the AV slightly touched the passing human driver and the other was in the scenario RTSP-RD when the AV entered the designed conflict zone promptly behind the human driver.

Risk

For the objective risk evaluation and as risk feature in the automated vehicle, we use the PET value. Positive values represent the AV passing before the human and negative values the opposite. In Figure 3 the distribution of the PET values is shown according to the aggressive and passive AV setup. The bars are semitransparent to display both AV setups in one graph. The gap after zero indicates that almost no situations appeared where the AV entered directly before the human driver. Exceptions are the two incidents described before and one appearance of a PET value slightly above 0.5 seconds in the aggressive AV setup. In the majority of cases, the AV yielded and entered the conflict zone after the drivers.

Figure 3: Histogram of the PET values for both AV setups (semitransparent bars)

The small negative PET values indicate that the AV entered the PET zone directly after the driver left which the driver was not able to percept in our setup (missing mirrors in the simulation). So, this should not have influenced the subjective risk assessment but seems not desired.

The subjective risk was assessed by the driver on a ten point scale after every run. Here, the drivers evaluated the riskiness of the behavior of the automated vehicle. The average risk values per scenario and AV setup are shown in Table 2. Below the mean values, there are the confidence intervals on a confidence level of 95%. As expected, the mean risk for the aggressive setup is higher than for the passive setup. A main factor of influence is the change in the situation which was not controlled in the simulation design. So, there is a difference in the rating when the AV enters before the driver compared to when it yields and stops at the intersection. Latter is usually considered less risky and scores lower on the scale accordingly. The low value for RTSP-RD with a passive AV is related to this issue. There are only 14 "AV first" situations (29%) in this scenario. This could explain the low risk rating. Highest rating is SAP-RD with a score of 4.9. Here, there are 36 "AV first" occurrences (75%) which is increasing the risk score noticeably. The dependency on the situational outcome is further investigated by comparing the risk score according to the situation and the AV setup (see Figure 4). Here, an unbalanced two-way ANOVA was performed showing a main effect in the AV setup (F(1, 570) = 20.79, p < .001) as well as in the outcome of the situation (F(1, 570) = 33.79, p < .001). Overall, the risk values remain in a lower part of the scale. For investigating the relationship between our objective risk measure and the risk score, a correlation analysis was performed. We took only the samples with positive PET (AV first), since the driver could not perceive vehicles passing behind the car. There was only a weak correlation (r= -0.42). A closer look into the data of some test drivers revealed that there might have been a misunderstanding of the risk score.

Naturalness

After each passing of the intersection, the driver was asked to rate the naturalness of the behavior of the present vehicle on a 1-10 scale. It was not mentioned that the present vehicle is controlled automatically. The mean score over all crossings was considerably high with 7.3 (sd= 2.56).

Further, the score was evaluated for each of the two AV setups (aggressive vs. passive) and separately for each of the six scenarios. The results are shown in Figure 5. Here, the mean scores and their confidence intervals are displayed. It shows that the scores for the aggressive setup are overall lower than for the passive setup. The LTAP-OD scenario stands out with a very low average of 5.9 (CI= .75). High scores are reached for the passive AV setup in the LTSP-LD scenario (mean= 8.4; CI= .56) and in the RTSP-RD scenario (mean= 8.6; CI= .51).

Figure 5: Naturalness score per scenario with both AV setups

The high scores seem to be related to the numbers of the AV passing before the vehicle. The outcome of the situation and both AV setups is compared and shown in Figure 6. An unbalanced two-way ANOVA shows that there is a main effect in the AV setup (F(1, 570) = 12.9, p < .001). Accordingly, the passive vehicle setup is rated more natural than the aggressive one. These results suggest that in the ranking there is a relationship between the risk and the naturalness ranking by the human drivers. This is confirmed by testing the correlation of all sequences between both scores (r= -0.66). However, the objective risk in form of PET shows a weak correlation with the naturalness score for positive PET values (r= 0.24).

Discussion and Conclusion

The objective risk evaluation using PET showed two dangerous incidents where the automated vehicle entered too close behind the human driver and even collided once. We identified this weakness of our approach which also leads to the high rate of very low negative PET values in Figure 3. Since the decision-making algorithm is only using the predicted position of the other vehicle to determine the risk of a possible trajectory, the vehicle remains unconsidered as soon as it exits the conflict zone. Basically, a memory attribute is required for the AV to realize that the conflict zone was recently occupied and shall not be entered before a safety margin. The subjective risk evaluation indicated that both AV setups are distinguishable and the aggressive setup is assessed as more risky in its behavior. We also showed that this is mainly related to the higher numbers of situations where the AV enters the intersection first. The weak correlation between the objective and subjective risk measures shows that the subjective evaluation is reasonable and extends the existing validation. Further, there might be another risk measure that is more related to the human perception which could be investigated in future research. The overall naturalness score was in the upper part of the scale. In the scenario comparison, the LTAP-OD was rated less natural which is explainable by the missing visualization of blinkers in the simulation. So, the drivers were confused of the vehicle stopped on the opposite site of the intersection or in the other situation turning left and passing in front of them. Apart from that, the high scores for passive AVs in the two scenarios where both vehicles go in the same direction after passing the intersection are interesting. There is most likely a relation to the amounts of situations where the AV yields rather than passes in front. Showing the correlation between naturalness and the subjective risk score, indicates that the human driver assessed less risky behavior as more natural. This is rather unexpected since the human behavior can intentionally or unintentionally show very risky patterns. In further works, we will investigate other features and can compare the scores by showing the same scenes to other drivers for a cross-validation.

Finally, the study showed the relevance of a subjective evaluation of automated vehicle behavior and provides a method to compare automated vehicle algorithm designs and setups. This is especially reasonable when a human like behavior is desired in the automated vehicle control. In our perspective, there are several advantages to aim for this goal since behavior models for the prediction within automated vehicles are commonly designed using normal driving data in a human-driver-only domain. The changes in the behavior that might appear when an automated vehicle is identified remain unconsidered and seem challenging to be integrated. So, an automated vehicle capable of blending in the current traffic environment seems more desirable.

Figure 1 :

 1 Figure 1: bird-view (left) and in-vehicle perspective (right) of the simulation tool

Figure 2 :

 2 Figure 2: Scenarios grouped by direction of the AV (green) relative to the human driver (blue)

Figure 4 :

 4 Figure 4: Risk score AV first vs. driver first with both AV setups

Figure 6 :

 6 Figure 6: Naturalness score AV first vs. driver first with both AV setup

Table 1 : Order of passing according to AV setup

 1

	AV setup	AV first	driver first
	Aggressive	140	146
	Passive	95	193

Table 2 : Mean risk scores per scenario in dependence of AV setup

 2

	AV setup		LTAP-RD LTAP-OD LTSP-LD	SAP-RD	SAP-LD	RTSP-RD
	aggressive	Mean	3.4	4.3	4.4	4.9	4.4	3.1
	aggressive	CI	0.66	0.77	0.91	0.95	0.77	0.71
	passive	Mean	2.9	2.5	2.9	3.7	3.3	1.9
	passive	CI	0.78	0.49	0.75	0.88	0.63	0.38

Acknowledgment

This work was supported by Valeo Driving Assistance Research in cooperation with the Valeo research team led by Dr. Benazouz Bradai and project leader Paulo Resende.


René Henrion

Alberto Seeger

CONDITION NUMBER AND ECCENTRICITY OF A CLOSED CONVEX CONE

We discuss some extremality issues concerning the circumradius, the inradius, and the condition number of a closed convex cone in R n . The condition number refers to the ratio between the circumradius and the inradius. We also study the eccentricity of a closed convex cone, which is a coefficient that measures to which extent the circumcenter differs from the incenter.

Introduction

This paper is the last part of a triptych initiated in [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF] and continued in [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF]. It deals with two concepts related to the geometric nature of a convex cone: the circumradius and the inradius. A few words on notation are in order before we recall these concepts. In the sequel the symbol n indicates the set of nontrivial closed convex cones in the Euclidean space R n . That a convex cone is nontrivial means that it is different from the singleton {0} and different from the whole space R n . For avoiding trivialities we assume that the dimension n is at least three. Some special subsets of n play a prominent role in the discussion, namely

sol n = {K ∈ n : K is solid}, ptd n = {K ∈ n : K is pointed}, reg n = {K ∈ n : K is regular}.
Recall that a closed convex cone is solid if its topological interior is nonempty, and it is pointed if it contains no line. Regularity is understood as the combination of solidity and pointedness.

The inradius of K ∈ n is defined as the coefficient

(1) ρ(K) = sup x∈K∩S n dist[x, ∂K],
where S n is the unit sphere of R n , ∂K is the boundary of K, and dist[ • , ] stands for the distance function to a set . Various interpretations of ρ(K) and calculus rules for computing this coefficient have been proposed in [START_REF] Epelman | A new condition measure, preconditioners, and relations between different measures of conditioning for conic linear systems[END_REF], [START_REF] Freund | On the primal-dual geometry of level sets in linear and conic optimization[END_REF], [START_REF] Freund | Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm[END_REF], [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF], [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF], [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF] and other places. We recall that (1) is a matter of finding the radius and center of a largest ball contained in K:

(2)

maximize r x = 1 r ∈ [0, 1] x + rB n ⊂ K,
where B n denotes the closed unit ball of R n . The coefficient ρ(K) is equal to the optimal value of the maximization problem [START_REF] Barker | Self-dual cones in Euclidean spaces[END_REF]. The function ρ : n → [0, 1] is continuous if n is equipped with the gap metric

δ(K 1 , K 2 ) = max max x∈K 1 ∩S n dist[x, K 2 ], max x∈K 2 ∩S n dist[x, K 1 ] ,
or with any other equivalent metric for that matter (cf. [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF]Proposition 6.3]).

We mention in passing that convergence with respect to the metric δ is equivalent to convergence in the Painlevé-Kuratowski sense (cf. [START_REF] Iusem | Distances between closed convex cones: old and new results[END_REF]Proposition 4.4]). This fact will be used on several occasions. If K ∈ n is solid, then the solution set inc (K) = {x ∈ K ∩ S n : dist[x, ∂K] = ρ(K)}

to the variational problem (1) is a singleton. The unique element of this set is denoted by π inc (K) and called the incenter of K. Theorem 2.9 of [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF] asserts that π inc : sol n → R n is a continuous map. The circumradius of K ∈ n , denoted by μ(K), is defined as the optimal value of the minimization problem

(3) minimize r x = 1 r ∈ [0, 1] K ⊂ M(x, r).
Here M(x, r) stands for the closed convex cone generated by the ball x + rB n , that is,

M(x, r) = cl α≥0 α(x + rB n ) .
The closure operation "cl" is superfluous when

r = 1. If K ∈ n is pointed, then circ (K) = {x ∈ R n : (x, r) solves (3)}
is a singleton. The unique element of this set is denoted by π circ (K) and called the circumcenter of K. By combining Proposition 4.8 and Theorem 5.2 in [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF], one sees that μ : n → [0, 1] and π circ :

ptd n → R n are continuous functions. This is all what the reader needs to know for having a good understanding of our work. The organization of the paper is as follows. Section 2 discusses some extremality issues concerning inradii and circumradii. Inspired by the definition of the condition number of a nonsingular matrix, we refer to the ratio

c(K) = μ(K) ρ(K)
as the condition number of a solid cone K ∈ n . The analysis of this concept is the object of Section 3. A regular cone

K ∈ n is non-eccentric if π inc (K) = π circ (K)
, otherwise it is said to be eccentric. The eccentricity is a coefficient that measures the gap between the incenter and the circumcenter:

(4) e(K) = π inc (K) -π circ (K) .
This coefficient is studied in detail in Section 4.

Extremality issues for inradii and circumradii

Preliminary results

Inradii and circumradii are dual objects. Indeed, for all K ∈ n one has

μ(K) = 1 -[ρ(K + )] 2 , (5) ρ(K) = 1 -[μ(K + )] 2 ,
where K + stands for the dual cone of K, i.e., [START_REF] Goffin | The relaxation method for solving systems of linear inequalities[END_REF] circ (K) = inc (K + ). [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF] These duality relationships have been established in [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF]Theorem 5.2]. Of course, the inequality

K + = {y ∈ R n : y, x ≥ 0 for all x ∈ K}. Furthermore, inc (K) = circ (K + ),
(8) ρ(K) ≤ μ(K)
holds for any K ∈ n . The next lemma is easy and consistent with intuition. We mention it only for the sake of subsequent use. Recall that a revolution cone is a set of the form

(y, θ) := x ∈ R n : y, x ≥ x cos θ ,
where y is a unit vector of R n . The parameter θ ∈ [0, π/2] is referred to as the half-aperture angle of the cone.

Lemma 2.1. For K ∈ n the following conditions are equivalent: Section 3.1]. By the way, the equivalence between ball-generated cones and revolution cones holds even in Hilbert spaces. In fact, by combining Lemmas 4.12 and 5.1 in [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF] one gets

(a) ρ(K) = μ(K). (b) K is a ball-generated cone. (c) K is a revolution cone. Proof. That (b) ⇔ (c) is mentioned in [6,
(y, θ) = M(y, sin θ), M(x, r) = (x, arcsin r).
For proving (b) ⇒ (a) one just needs to observe that [START_REF] Iusem | Pointedness, connectedness, and convergence results in the space of closed convex cones[END_REF] ρ(M(x, r)) = μ(M(x, r)) = r for all (x, r) ∈ S n × [0, 1]. The proof of (a) ⇒ (b) is more subtle. Let r := ρ(K) = μ(K). We suppose that r > 0, otherwise K is a ray, i.e., a cone generated by a ball of radius 0. Let x := π inc (K) and w ∈ circ (K). In such a case [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF] x + rB n ⊂ K ⊂ M( w, r).

Since ρ(M( w, r)) = r, the unit vector x must be the incenter of M( w, r). In other words,

x = π inc (M( w, r)) = w.
Hence, the chain of inclusions in [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF] yields M( x, r) = K = M( w, r).

Remark 2.2. It is well known that the dual of a revolution cone is a revolution cone. Hence, the dual of a ball-generated cone is a ball-generated cone. More precisely, [M(x, r)]

+ = M x, 1 -r 2 .
For the reader's convenience we recall below a technical result (cf. [8, Lemma 2.25]) that characterizes the incenter of a solid polyhedral cone. Such result will be used on a few occasions. The notation "pos" stands for positive (or convex conic) hull.

Lemma 2.3. Let K ∈ n be a solid polyhedral cone represented by [START_REF] Iusem | Measuring the degree of pointedness of a closed convex cone: a metric approach[END_REF] x ∈ pos {f i : i ∈ I ( x)} , [START_REF] Iusem | Computing the radius of pointedness of a convex cone[END_REF] where I ( x) is the set of indices j ∈ {1, . . . , m} such that

K = x ∈ R n : f 1 , x ≥ 0, . . . , f m , x ≥ 0 , where {f i } m i=1 is a finite collection of unit vectors in R n . Then x = π inc (K) if and only if x ∈ K ∩ S n ,
(13) f j , x = min 1≤i≤m f i , x .
Furthermore, if x is the incenter of K, then the minimum in (13) is equal to ρ(K).

As first use of Lemma 2.3 we derive an explicit formula for computing the incenter of a special type of polyhedral cone. Recall that a simplicial cone in R n is a polyhedral cone generated by a basis of R n . In other words, K ∈ n is simplicial if and only if

(14) K = {Gλ : λ ∈ R n + }
with G standing for a nonsingular matrix of order n. A simplicial cone in R n is regular and has exactly n facets. General information on simplicial cones and facial analysis can be found in [START_REF] Barker | Theory of cones[END_REF].

Theorem 2.4. Suppose that K ∈ n is generated by the columns of a nonsingular matrix G = [g 1 , . . . , g n ]. Let f j denote the j -th column of F = (G -1 ) T . Consider the following statements:

(a) g i , g j ≥ 0 for all i, j ∈ {1, . . . , n}.

(b) n j =1 g i , g j f j ≥ 0 for all i ∈ {1, . . . , n}. (c) The incenter of K is equidistant to each facet of K. (d)
The incenter of K is equal to the vector [START_REF] Kelly | CP-rays in simplicial cones[END_REF] x :=

n i=1 f i g i n i=1 f i g i . (e) ρ (K) = n i=1 f i g i -1 .
Then one has (a) ⇒ (b) ⇔ (c) ⇔ (d) ⇒ (e).

Proof. Before starting with the proof itself, observe that

(16) f j , g i = δ ij for all i, j ∈ {1, . . . , n} ,
where δ ij refers to the Kronecker delta. Hence, the simplicial cone ( 14) can be rewritten as

K = x ∈ R n : f1 , x ≥ 0, . . . , fn , x ≥ 0 with f j = f j -1 f j .
For convenience we split the proof in several parts: (a) ⇒ (b). This implication is obvious. (b) ⇔ (d). The vector x given by ( 15) clearly belongs to K ∩ S n . Thanks to [START_REF] Seeger | Epigraphical cones I[END_REF] one has

(17) f j , x = κ := n i=1 f i g i -1
for all j ∈ {1, . . . , n}. Hence,

I ( x) := j ∈ {1, . . . , n} : f j , x = min 1≤i≤n fi , x
is equal to the whole index set {1, . . . , n}. By Lemma 2.3 one has

x = π inc (K) ⇔ x ∈ pos f1 , . . . , fn ⇔ g 1 , x ≥ 0, . . . , g n , x ≥ 0 ⇔ n j =1
g i , g j f j ≥ 0 for all i ∈ {1, . . . , n}.

(d) ⇒ (e). It follows from [START_REF] Seeger | Epigraphical cones II[END_REF] and the last part of Lemma 2.3.

(d) ⇒ (c). The facets of K are F j = K ∩ f ⊥ j with f ⊥ j = {x ∈ R n : f j , x = 0}.
By assumption the vector x given by ( 15) is the incenter of K. The condition [START_REF] Seeger | Epigraphical cones II[END_REF] shows that x is equidistant to each hyperplane f ⊥ j , the common distance being ρ(K). Let ξ j denote the orthogonal projection of x into f ⊥ j . Hence,

ξ j = x -f j , x f j = x -ρ(K) f j and ξ j -x = ρ(K). It follows that ξ j ∈ F j and dist[ x, F j ] = dist[ x, f ⊥ j ] = ρ(K).
This proves the condition (c).

(c) ⇒ (d). One has ū := π inc (K) = n i=1 λ i g i , where each λ i is positive. Note that

ρ(K) = dist[ ū, ∂K] = min 1≤j ≤n f j , ū f j = min 1≤j ≤n λ j f j ,
where the last equality is due to [START_REF] Seeger | Epigraphical cones I[END_REF]. Hence,

(18) ρ(K) f j ≤ λ j
for all j ∈ {1, . . . , n}. But,

ρ(K) = dist[ ū, F j ] ≥ dist[ ū, f ⊥ j ] = f j , ū f j = λ j f j ,
where the first equality is due to the assumption (c). So, ( 18) is in fact an equality, and therefore

ū = ρ(K) n i=1 f i g i .
A due normalization shows that ū is equal to the vector given by [START_REF] Kelly | CP-rays in simplicial cones[END_REF].

The condition (a) says that the angle between any pair of generators of K does not exceed π/2. This requirement is stronger than (b). To see this, consider the simplicial cone K generated by the columns of

G = 1 -1 -1 0 1 -2 0 0 1 .
The condition (a) is clearly violated. However

F = (G -1 ) T = 1 0 0 1 1 0 3 2 1 , f 1 = √ 11, f 2 = √ 5, f 3 = 1,
and a direct computation shows that (b) holds. Thus, one can use [START_REF] Kelly | CP-rays in simplicial cones[END_REF] for computing the incenter of K. Kelly et al. [START_REF] Kelly | CP-rays in simplicial cones[END_REF] propose a different concept of "center" for a simplicial cone K, namely, any point in the interior of K which is equidistant from each facetal hyperplane. This property is satisfied by the vector x given by [START_REF] Kelly | CP-rays in simplicial cones[END_REF]. However, such x may not be the incenter of K. To see this, consider the simplicial cone K generated by the columns of

G = 1 -1 -1 0 1 1 0 0 1 . One has F = (G -1 ) T = 1 0 0 1 1 0 0 -1 1 , f 1 = √ 2, f 2 = √ 2, f 3 = 1,
and the condition (b) is violated. As a consequence, π inc (K) and x do not coincide. Theorem 2.4 can be dualized in order to obtain a formula for computing the circumcenter of a simplicial cone.

Corollary 2.5. Let K ∈ n be the simplicial cone generated by the columns of the nonsingular matrix G = [g 1 , . . . , g n ]. Let f j denote the j -th column of F = (G -1 ) T . Then the following conditions are equivalent:

(a)

n j =1 f i , f j g j ≥ 0 for all i ∈ {1, . . . , n}. (b)
The circumcenter of K is equal to the vector

ȳ := n i=1 g i f i n i=1 g i f i .
Proof. We apply Theorem 2.4 to K + and use the duality formula [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF].

Comparing the inradii of K and K +

The theory of solidity and pointedness indices for convex cones has been developed in recent years in [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF], [START_REF] Iusem | Measuring the degree of pointedness of a closed convex cone: a metric approach[END_REF], [START_REF] Iusem | Computing the radius of pointedness of a convex cone[END_REF], [START_REF] Iusem | Normality and modulability indices I: Convex cones in normed spaces[END_REF]. Within the context of such theory one can interpret the inradii of K and K + as follows:

ρ(K) = solidity index of K, ρ(K + ) = pointedness index of K.
The first question addressed in this section is to find a closed convex cone that is as solid and pointed as possible:

(19) maximize F (K) := ρ(K), ρ(K + ) with respect to K ∈ n .
Beware that solidity and pointedness are antagonic notions. What we mean by this is that both coefficients ρ(K) and ρ(K + ) cannot be large (i.e., near 1) at the same time. Proposition 2.6 characterizes the Pareto solutions to the above bicriteria optimization problem. By definition,

K ∈ n is a Pareto solution to (19) if there is no Q ∈ n such that ρ(K) ≤ ρ(Q) and ρ(K + ) ≤ ρ(Q + )
with at least one inequality being strict. Proposition 2.6. K ∈ n is a Pareto solution to [START_REF] Walkup | Continuity of some convex-cone-valued mappings[END_REF] if and only if K is a ball-generated cone.

Proof. As a consequence of ( 5) and ( 8) one gets

(20) [ρ(K)] 2 + ρ(K + ) 2 ≤ 1
for all K ∈ n . In fact, Corollary 8.4 in [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF] asserts something stronger: the image set

F ( n ) := ρ(K), ρ(K + ) : K ∈ n of the problem (19) is equal to = (r, t) ∈ R 2 + : r 2 + t 2 ≤ 1 .
Hence, K ∈ n is a Pareto solution to [START_REF] Walkup | Continuity of some convex-cone-valued mappings[END_REF] if and only if the pair ρ(K), ρ(K + ) lies in the upper right portion of , that is to say,

[ρ(K)] 2 + ρ(K + ) 2 = 1.
Lemma 2.1 and the duality formula (5) do the rest of the job.

Besides the inequality (20), is there any other interesting relationship between the inradii of K and K + ? The next result applies only to simplicial cones. That K ∈ n is orthogonal simply means that K a polyhedral cone generated by an orthogonal basis of R n . Every orthogonal cone is simplicial, but not conversely.

Theorem 2.7. If K ∈ n is simplicial, then (21) ρ(K)ρ(K + ) ≤ 1/n.
The above inequality becomes an equality if and only if K ∈ n is orthogonal.

Proof. Let K ∈ n be generated by the columns of a nonsingular matrix G = [g 1 , . . . , g n ] and let f j denote the j -th column of F = (G -1 ) T . There are vectors λ, β in R n + such that

ū := π inc (K) = Gλ = n i=1 λ i g i , v := π inc (K + ) = Fβ = n i=1 β j f j .
It follows that

ρ(K) = dist[ ū, ∂K] = min 1≤j ≤n f j , ū f j = min 1≤j ≤n λ j f j , ρ(K + ) = dist[ v, ∂K + ] = min 1≤i≤n g i , v g i = min 1≤i≤n β i g i .
Hence, for all j ∈ {1, . . . , n} one gets

ρ(K) f j ≤ λ j , ρ(K + ) g j ≤ β j ,
and therefore

(22) ρ(K)ρ(K + ) n j =1 f j g j ≤ n j =1 λ j β j .
Thanks to ( 16) and the Cauchy-Schwarz inequality, one has f j g j ≥ 1. Hence,

(23) n ≤ n j =1 f j g j .
For completing the proof of (21) it remains to observe that

(24) n i=1 λ j β j = G -1 ū, F -1 v = ū, (G -1 ) T F -1 v = ū, v is less than or equal to 1. If K ∈ n is orthogonal, then ρ(K) = ρ(K + ) = √ 1
/n, and ρ(K)ρ(K + ) = 1/n. Conversely, let K = pos{g 1 , . . . , g n } be a simplicial cone such that (21) holds as an equality. In such a case n j =1 f j g j = n, and therefore f j g j = 1 for all j ∈ {1, . . . , n}. This and ( 16) imply that, up to normalization, the sets {g j } n j =1 and {f j } n j =1 coincide. Hence,

g j g j , g i g i = f j f j , g i g i = f j , g i f j g i = 0
for all i = j , that is to say, K is orthogonal.

Remark 2.8. The product rule (21) does not apply beyond a simplicial context. For instance, the Lorentz (or ice-cream) cone

n = x ∈ R n : x 2 1 + • • • + x 2 n-1 1/2 ≤ x n satisfies ρ( n ) = ρ( + n ) = sin(π/4) = √ 1/2. So, ρ( n )ρ( + n ) = 1/2 is greater than 1/n.
We now derive a second product rule for inradii. It is less sharp than (21), but applies to arbitrary convex cones. One says that K ∈ n is Lorentzian if there exists an orthogonal matrix U of order n such that K = U( n ). Equivalently, a Lorentzian cone is a revolution cone with π/4 as half-aperture angle.

Proposition 2.9. For any K ∈ n one has

(25) ρ(K)ρ(K + ) ≤ 1/2.

The above inequality is an equality if and only if

K ∈ n is Lorentzian.
Proof. The proof of Proposition 2.6 shows that

(26) ρ(K)ρ(K + ) ≤ max (r,t)∈ rt = 1/2.
Note that (25) becomes an equality if K is Lorentzian. Indeed, in such a case one has

(27) ρ(K) = ρ(K + ) = 1/2.
Conversely, let K ∈ n be such that ( 25) is an equality. Then the pair (ρ(K), ρ(K + )) solves the maximization problem in (26). This is equivalent to saying that (27) holds. By combining (5) and Lemma 2.1 one deduces that K is generated by a ball. More precisely,

(28) K = M x, 1/2
with x = π inc (K). But, according to [START_REF] Goffin | The relaxation method for solving systems of linear inequalities[END_REF], the set (28) is equal to a revolution cone with x as revolution axis and arcsin( √ 1/2 ) = π/4 as half-aperture angle. So, up to orthogonal transformation, K is equal to n .

Inradii and maximal angles

The next theorem establishes a curious relationship between the circumradius μ(K) and the maximal angle

θ max (K) = max u,v∈K∩S n arccos u, v of K ∈ n .
It also establishes a link between the inradius ρ(K) and the maximal angle of the dual cone K + . Theorem 2.10. For all K ∈ n one has

[μ(K)] 2 ≤ 1 - 1 n 1 -cos [θ max (K)] , (29) [ρ(K)] 2 ≥ 1 n + 1 - 1 n cos[θ max (K + )]. (30)
Proof. In view of the duality formula [START_REF] Freund | Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm[END_REF], it is enough to prove the relation (30). Suppose that K ∈ n is solid, otherwise θ max (K + ) = π and (30) holds trivially. For convenience we distinguish between two cases. I. The polyhedral case. Let K be expressible as intersection of finitely many closed half-spaces, that is,

(31) K= x ∈ R n : f 1 , x ≥ 0, . . . , f m , x ≥ 0 .
Without loss of generality one assumes that (32) {f i } m i=1 is a positively independent collection of unit vectors of R n .

Note that the integer m could be much larger than n. According to Lemma 2.3, the incenter x = π inc (K) of the solid polyhedral cone (31) satisfies the condition ( 12) and

(33) f j , x = ρ(K)
for all j ∈ I ( x). The conic version of Caratheodory's theorem applied to [START_REF] Iusem | Computing the radius of pointedness of a convex cone[END_REF] yields the representation

(34) x = i∈I λ i f i ,
where each scalar λ i is positive and I is a subset of I ( x) such that {f i } i∈I is linearly independent. The general formulation and proof of the conic version of Caratheodory's theorem can be found in numerous references, see for instance Theorem 3.2 in [18, Chapter 1]. From (34) one gets

(35) f j , x = i∈I λ i f i , f j
for all j ∈ I , as well as

x, x = i∈I λ i f i , x = ρ(K) i∈I λ i .
Since x has unit length, it follows that

(36) i∈I λ i = 1/ρ(K).
Thanks to (33) and ( 35), for all j ∈ I one has

(37) ρ(K) = λ j + i∈I \{j } λ i f i , f j .
Since the f i are unit vectors in K + , one has

γ := cos[θ max (K + )] ≤ f i , f j .
Hence,

i∈I \{j } λ i f i , f j ≥ γ i∈I \{j } λ i = γ i∈I λ i -λ j
for all j ∈ I . This, together with (36) and (37), produces the inequality

ρ(K) ≥ λ j + γ 1 ρ(K) -λ j
for all j ∈ I . By passing to the sum and using (36) again, one obtains

card(I ) ρ(K) - γ ρ(K) ≥ 1 -γ ρ(K) .
But the cardinality of I cannot exceed n. Hence, the above line leads to

[ρ(K)] 2 -γ ≥ 1 -γ n ,
which is just another way of writing (30).

II. The nonpolyhedral case. Suppose that K is not polyhedral. Theorem 4.4 in [START_REF] Seeger | Epigraphical cones II[END_REF] asserts that any closed convex cone in an Euclidean space can be written as Painlevé-Kuratowski limit of a sequence of polyhedral cones. As a consequence of this approximation result, there exists a sequence {K ν } ν∈N of polyhedral cones K ν ∈ n such that lim ν→∞ δ(K ν , K) = 0.

But the celebrated Walkup-Wets Isometry Theorem (cf. [START_REF] Walkup | Continuity of some convex-cone-valued mappings[END_REF]Theorem 1]) says that the duality operation Q → Q + is an isometry on ( n , δ), i.e.,

δ(Q + 1 , Q + 2 ) = δ(Q 1 , Q 2 ) for all Q 1 , Q 2 ∈ n .
Hence, one also has lim

ν→∞ δ(K + ν , K + ) = 0.
As shown in Part I, for each ν ∈ N one can write

(38) [ρ(K ν )] 2 ≥ 1 n + 1 - 1 n cos[θ max (K + ν )].
Thanks to the continuity of ρ : n → [0, 1] and θ max :

n :→ [0, π], one has lim ν→∞ ρ(K ν ) = ρ(K), lim ν→∞ θ max (K + ν ) = θ max (K + ).
It suffices then to pass to the limit in (38) as ν → ∞.

There are a number of interesting consequences that can be derived from Theorem 2.10. Recall that K ∈ n is said to be ⎧ ⎨ ⎩

supradual if K ⊃ K + , infradual if K ⊂ K + , selfdual if K = K + .
Clearly, infraduality implies pointedness and supraduality implies solidity. A quantitative version of these statements reads as follows.

Corollary 2.11. One has:

(a) If K ∈ n is infradual, then μ(K) ≤ √ 1 -(1/n). (b) If K ∈ n is supradual, then ρ(K) ≥ √ 1/n. Proof. K ∈ n is infradual if and only if θ max (K) ≤ π/2. Part (a)
is then a consequence of (29). Similarly, that K ∈ n is supradual is equivalent to saying that θ max (K + ) ≤ π/2. Hence, part (b) is a consequence of (30).

The lower bound in Corollary 2.11(b) is optimal. Indeed, (39) min

K∈ n K supradual ρ(K) = 1/n
with attainment of the minimum if K ∈ n is orthogonal. Similarly, the upper bound in Corollary 2.11(a) is optimal because (40) max

K∈ n K infradual μ(K) = 1 -(1/n)
with attainment of the maximum at any K ∈ n that is orthogonal. Remark 2.12. Every orthogonal cone is selfdual. Beware that an arbitrary selfdual cone may not achieve the minimum in (39) or the maximum in (40). To see this, consider the cone K generated by the vectors

1 1 1 , 0 1 1 , -1 0 1 , 0 -1 1 , 1 -1 1 .
This cone is proposed by Barker and Foran [START_REF] Barker | Self-dual cones in Euclidean spaces[END_REF] as example of self-dual polyhedral cone that is not simplicial. On the other hand, it is shown in [8, Example 2.3] that ρ(K) > √ 1/3, which means that K does not achieve the minimum in (39).

For the sake of completeness we mention another result in the same vein as Corollary 2.11.

Corollary 2.13. Let K ∈ n . (a) If K is infradual, then ρ(K) ≤ √ 1/2. (b) If K is simplicial and infradual, then ρ(K) ≤ √ 1/n. (c) If K is supradual, then μ(K) ≥ √ 1/2. (d) If K is simplicial and supradual, then μ(K) ≥ √ 1 -(1/n).
Proof. Part (a) is a consequence of ( 5) and [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF]. Part (b) follows from Theorem 2.7. Parts (c) and (d) are obtained by applying (a) and (b) to the dual cone K + .

Condition number of a convex cone

As mentioned before, the condition number of a solid cone K ∈ n is defined as the ratio c(K) = μ(K)/ρ(K). By mimicking the parlance of numerical linear algebra, one says that K is well-conditioned if c(K) is near to 1 and ill-conditioned if c(K) is much larger than 1.

Example 3.1. Consider the elliptic cone

E A = (z, t) ∈ R n-1 × R : z, Az ≤ t
associated to a positive definite symmetric matrix A of order n -1. Proposition 6.4 in [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF] asserts that

ρ(E A ) = [1 + λ max (A)] -1/2 , μ(E A ) = [1 + λ min (A)] -1/2 ,
where λ max (A) and λ min (A) denote, respectively, the largest and smallest eigenvalue of A. Hence,

c(E A ) = 1 + λ max (A) 1 + λ min (A) .
Note that c(E A ) = 1 if and only if A is a positive multiple of the identity matrix.

Example 3.2. As a generalization of the above example, consider the

epigraph epi φ = {(z, t) ∈ R n-1 × R : φ(z) ≤ t} of a norm φ on R n-1 .
Such set is clearly a regular cone in R n . Lemmas 3.1 and 4.1 in [START_REF] Seeger | Epigraphical cones I[END_REF] assert, respectively, that

ρ(epi φ) = 1 + β 2 φ -1/2 , μ(epi φ) = α φ 1 + α 2 φ -1/2 ,
where

α φ := min z =1 φ(z) and β φ := max z =1 φ(z).
Hence,

c(epiφ) = 1 + β 2 φ 1 + α 2 φ .
Note that c(epiφ) = 1 if and only if φ is constant on the unit sphere S n-1 .

What does it mean actually that c(K) is near to 1? The next theorem provides an answer to this question. We establish first a topological lemma concerning the collection ball n = {M(x, r) : (x, r) ∈ S n × [0, 1]} of ball-generated cones in R n . Lemma 3.3. ball n is a closed set in the metric space ( n , δ). Proof. The lemma is surely known, so we give only a sketch of the proof. Let {K ν } ν∈N be a sequence in ball n such that lim ν→∞ δ(K ν , K) = 0. For each ν ∈ N one has K ν = M(x ν , r ν ) with (x ν , r ν ) ∈ S n × [0, 1]. By taking a subsequence if necessary, one may suppose that Then

lim ν→∞ c(K ν ) = 1 ⇐⇒ lim ν→∞ dist K ν , ball n = 0.
Proof. Let us start with the "if" part. Suppose that

d ν := dist K ν , ball n goes to 0 as ν → ∞. For each ν ∈ N, pick (x ν , r ν ) ∈ S n × [0, 1] such that M ν = M(x ν , r ν ) achieves the distance from K ν to ball n , that is, (43) δ (K ν , M ν ) = d ν .
As shown in [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF]Proposition 6.3], the inradius ρ is a nonexpansive function on ( n , δ). Hence,

(44) |ρ(K ν ) -ρ(M ν )| ≤ δ (K ν , M ν ) .
By combining ( 9), (43), and (44), one gets

(45) ρ ( K ν ) = ρ(M ν ) + ε ν = r ν + ε ν
with {ε ν } ν∈N converging to 0. We now examine the term μ(K ν ). The nonexpansiveness of ρ and the Walkup-Wets Isometry Theorem yield

ρ(K + ν ) -ρ(M + ν ) ≤ δ K + ν , M + ν = δ (K ν , M ν ) . But M + ν = [M(x ν , r ν )] + = M x ν , 1 -r 2 ν .
One gets in this way

ρ(K + ν ) = ρ(M + ν ) + γ ν = 1 -r 2 ν + γ ν
with {γ ν } ν∈N converging to 0. Thanks to the duality relation [START_REF] Freund | Condition-based complexity of convex optimization in conic linear form via the ellipsoid algorithm[END_REF], one arrives at

μ(K ν ) = 1 -ρ(K + ν ) 2 1/2 = 1 - 1 -r 2 ν + γ ν 2 1/2 .
We must show that

(46) c(K ν ) = 1 -1 -r 2 ν + γ ν 2 1/2 r ν + ε ν
goes to 1 as ν → ∞. In view of (45), the Uniform Solidity Condition (42) implies that the sequence {r ν } ν∈N remains away from 0. Hence, the numerator and denominator of the quotient (46) are asymptotically equal. More precisely, both behave as r ν . We now prove the "only if" part. Suppose that lim ν→∞ c(K ν ) = 1. We claim that the upper limit Since {K ϕ(ν) } ν∈N lies in the compact metric space ( n , δ), there exists yet another increasing function ψ : N → N and an element K ∈ n such that

lim ν→∞ δ(Q ν , K) = 0 with Q ν = K ϕ(ψ(ν)) . Let r ν = ρ(Q ν ) and s ν = μ(Q ν ). Then M(x ν , r ν ) ⊂ Q ν ⊂ M(w ν , s ν ) with x ν = π inc (Q ν ) and w ν ∈ circ (Q ν ). Since ρ, μ are continuous functions on n , one gets r := lim ν→∞ r ν = ρ( K) s := lim ν→∞ s ν = μ( K).
The Uniform Solidity Condition (42) implies that r > 0. Hence, K is solid and

1 = lim ν→∞ c(K ν ) = lim ν→∞ c(Q ν ) = lim ν→∞ s ν /r ν = s/r,
that is to say, s = r. Given that {x ν } ν∈N and {w ν } ν∈N are in the compact set S n , there is an increasing function φ : N → N such that both limits

x = lim ν→∞ x φ(ν) , w = lim ν→∞ w φ(ν)
exist. By continuity arguments one obtains x = π inc ( K) and w ∈ circ ( K). Now, passing to Painlevé-Kuratowski limits in the sandwich

M(x φ(ν) , r φ(ν) ) ⊂ Q φ(ν) ⊂ M(w φ(ν) , s φ(ν) ),
and keeping in mind that s = r, one arrives at

x + r B n ⊂ M( x, r) ⊂ K ⊂ M( w, r).
We are in the same situation as in [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF], so one deduces that K = M( w, r) is a ball-generated cone. Hence,

= lim ν→∞ dist K ϕ(ν) , ball n = lim ν→∞ dist Q ν , ball n = lim ν→∞ dist K, ball n = 0.
This proves our claim and completes the proof of the theorem.

Eccentricity of a regular cone

The eccentricity of a regular cone K ∈ n is defined by the expression (4), that is to say, it is the gap between the incenter and the circumcenter of K. By squaring both sides of (4) and keeping in mind that π inc (K) and π circ (K) are unit vectors, one gets

[e(K)] 2 = π inc (K) -π circ (K) 2 = 2 -2a(K),
where a(K) := π inc (K), π circ (K) . In other words, the eccentricity

(47) e(K) = 2(1 -a(K))
of a regular cone K has to do also with the angle formed by π inc (K) and π circ (K).

Figure 1. Mutually dual simplicial cones in R 3 . One ray corresponds to the incenter of K, the other ray corresponds to the circumcenter of K (i.e., the incenter of K + ). In this example the angle between π inc (K) and π circ (K) is almost π/4, suggesting a rather high degree of eccentricity. 6) and [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF]. The strict inequality in (b) follows from (47) and the fact that a(K) > 0 for all K ∈ reg n . Recall that π inc (K) belongs to the interior of K and π circ (K) belongs to the interior of K + . The set in (c) is an interval because reg n is arc-connected (cf. [START_REF] Iusem | Pointedness, connectedness, and convergence results in the space of closed convex cones[END_REF]Proposition 7.3]).

The next theorem provides an upper bound for the eccentricity in terms of the coefficient (K) = ρ(K) + ρ(K + ) 1 + ρ(K)ρ(K + ) .

Such an expression is well defined for all K ∈ n and satisfies (48) 0 ≤ max ρ(K), ρ(K + ) ≤ (K) ≤ 1.

These inequalities are all strict when K is regular. Proof. We claim that the inner product of x = π inc (K) and y = π circ (K) is greater than (K). We suppose that x = y, otherwise we are done. If one sets r = ρ(K) and s = μ(K), then one can write x + rB n ⊂ K ⊂ M(y, s).

Hence, for all u ∈ B n one has x + ru ∈ M(y, s) or, equivalently, (50) t x + ru ≤ y, x + ru with t = √ 1s 2 = ρ(K + ). We exploit the relation (50) for the particular choice u = -y. Since x and y are not collinear, one has xry > 1r x, y , and therefore t (1r x, y ) < x, yr.

After simplification one obtains

x, y > r + t 1 + rt .

Though we know that the eccentricity of a regular cone is smaller that √ 2, it remains an open question to compute

E n = sup{e(K) : K ∈ reg n }.
It is not clear whether this supremum depends on n and which one is its exact value. For the sake of illustration we give below an example showing that

(55) 2 - √ 2 ≤ E 3 ≤ √ 2.
Example 4.5. For each ν ≥ 2, let K ν be the simplicial cone generated by the columns of

G ν = ⎡ ⎣ 0 ν -1 ν -1 0 √ 1 -ν -2 0 1 0 √ 1 -ν -2 ⎤ ⎦ .
A long and tedious computation shows that lim ν→∞ e(K ν ) = 2 -√ 2, which explains the lower bound in (55).

  lim ν→∞ (x ν , r ν ) = ( x, r) ∈ S n × [0, 1]. A routinary work shows that lim sup ν→∞M(x ν , r ν ) ⊂ x, r) ⊂ lim inf ν→∞ M(x ν , r ν ),where the upper and lower limits are understood in the Painlevé-Kuratowski sense. One proves in this way that K = M( x, r). Hence, K ∈ ball n . In view of Lemma 2.1, a solid cone K ∈ n satisfies c(K) = 1 if and only if K ∈ ball n . An asymptotic version of this statement is formulated in the next theorem. The notation from K ∈ n to the closed set ball n . Since the metric space ( n , δ) is compact (cf. [9, Proposition 2.1]), the subset ball n is compact as well. This explains why the minimum in (41) is attained. Theorem 3.4. Let {K ν } ν∈N be a sequence in n satisfying the Uniform Solidity Condition (42) inf ν∈N ρ(K ν ) > 0.

=

  lim sup ν→∞ dist K ν , ball n is equal to zero. Let ϕ : N → N be an increasing function such that lim ν→∞ dist K ϕ(ν) , ball n = .

Proposition 4 . 1 .

 41 The function e : reg n → R is continuous and satisfies the following properties: (a) e(K + ) = e(K) for all K ∈ reg n . (b) 0 ≤ e(K) < √ 2 for all K ∈ reg n . (c) {e(K) : K ∈ reg n } is an interval. Proof. Both functions π inc and π circ are continuous on reg n . Part (a) is a consequence of (

Theorem 4 . 2 .such that 2 ( 1 -

 4221 For all K ∈ reg n one has(49) e(K) < 2 (1 -(K)).Furthermore, there exists sequence{K ν } ν∈N in reg n (K ν ))e(K ν ) → 0 as ν → ∞.

This confirms our claim and completes the proof of (49). For proving the last part of the theorem we consider a revolution cone

whose revolution axis is a given vector z ∈ S n . For each ν ≥ 1 one has

and therefore e(K ν ) = 0. On the other hand,

for all K ∈ reg n . However, these upper bounds are less sharp than (49). 53) is a consequence of (52).

The following theorem proposes an alternative to the upper bound (49), but it concerns only the class of simplicial cones. Note that the new bound (54) is exact for orthogonal cones, whereas (49) is not.

Proof. Consider again the proof of Theorem 2.7. By combining ( 22), (23), and (24), one gets

This yields the announced relation (54).