Alberto Seeger
email: alberto.seeger@univ-avignon.fr

Epigraphical cones II

Keywords: Mathematics Subject Classification: 46B10, 46B20, 52A41 Convex cone, epigraphical cone, sublinear function, smoothness, rotundity, Vinberg characteristic function, conic programming

Introduction

This is the second part of a work initiated in [START_REF] Seeger | Epigraphical cones I[END_REF] and whose aim is to survey the class of epigraphical cones. A convex cone in the Euclidean space R n+1 is an epigraphical cone if it can be represented as epigraph epif = {(x, t) ∈ R n+1 : f (x) ≤ t} of a nonnegative sublinear function f : R n → R. An epigraphical cone is always closed and nontrivial, i.e., different from the whole space and different from the zero cone. It is also solid in the sense that it has a nonempty topological interior. If K is an epigraphical cone in R n+1 , then its associated nonnegative sublinear function is given by

f K (x) = min{t ∈ R : (x, t) ∈ K}.
Any geometric statement on K can be formulated in terms of a corresponding analytic property of f K . The reference [START_REF] Seeger | Epigraphical cones I[END_REF] provides various examples of interesting epigraphical cones and explains how to combine them in order to produce new epigraphical cones. The next lemma is a bridge for passing from the class of epigraphical cones to the wider class of solid closed convex cones. We start by recalling a useful definition.

K = U (epif ). (1)
Proof. That a solid cone K admits the representation (1) has been proven in [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF]Proposition 2.8].

Conversely, that a cone of the form (1) is solid is a consequence of [21, Proposition 2.1] and the fact that solidity is a property that is invariant under orthogonal transformations.

Example 1.3. The nonnegative orthant R n+1 + is a solid closed convex cone in R n+1 . It is not an epigraphical cone, but it is orthogonally equivalent to an epigraphical cone. Consider the orthogonal matrix U = [u 1 , . . . , u n , u n+1 ] constructed as follows: the columns of the submatrix Ũ = [u 1 , . . . , u n ] form an orthonormal basis of the linear subspace

L n = {v ∈ R n+1 : v 1 + . . . + v n+1 = 0}
and u n+1 = (n + 1) -1/2 (1, . . . , 1) T is a unit vector orthogonal to L n . One can check that

R n+1 + = U (epif ),
where f : R n → R is the nonnegative sublinear function given by

f (x) = max 1≤i≤n+1 - √ n + 1 ( Ũ x) i
and ( Ũ x) i denotes the i-th component of Ũ x. One sees that f is nonnegative because Ũ x ∈ L n for all x ∈ R n . The set epif = {(x, t) ∈ R n+1 : ( Ũ x) i + (n + 1) -1/2 t ≥ 0 for all i = 1, . . . , n + 1} looks more involved than R n+1 + , but it has the merit of being an epigraphical cone.

The equality [START_REF] Aubin | Set-Valued Analysis[END_REF] expresses the fact that K is orthogonally equivalent to an epigraphical cone. The interest of having such a representation formula for K is clear: when it comes to study the structure of K, everything boils down to examining the function f . This observation is the leading motivation behind the present work. The organization of the paper is as follows:

-Section 2 concerns the facial structure of an epigraphical cone.

-Section 3 establishes a few results about rotundity and smoothness of epigraphical cones.

-Section 4 exploits the theory of epigraphical cones for obtaining some approximation results involving Painlevé-Kuratowski limits.

-Section 5 provides rules for computing the characteristic function of an epigraphical cone. The concept of characteristic function of a cone is understood in the sense of Vinberg [START_REF] Vinberg | The theory of homogeneous convex cones[END_REF].

-Section 6 deals with the application of epigraphical cones in optimization theory.

We keep the same notation and terminology as in [START_REF] Seeger | Epigraphical cones I[END_REF]. In particular, int(Ω), bd(Ω), cl(Ω) indicate respectively the interior, boundary, and closure of a set Ω. The unit sphere and the closed unit ball of R d are denoted by S d and B d , respectively. However, we deviate from the general spirit of [START_REF] Seeger | Epigraphical cones I[END_REF]. The emphasis now is put in the study of properties that are valid up to orthogonal characterizations, which allows to consider a class of convex cones larger than the class of epigraphical cones.

Facial analysis of epigraphical cones

A face of a nonempty closed convex set Ω in an Euclidean space is a nonempty subset F of Ω satisfying the following property:

(1 -λ)x + λy ∈ F with x, y ∈ Ω and λ ∈]0, 1[ =⇒ x, y ∈ F.
A face is necessarily closed and convex. If Ω is a closed convex cone, then so is every face of Ω. The next theorem tells how to identify the faces of an epigraphical cone. For the sake of clarity in the exposition, we assume that the associated nonnegative sublinear function f : R n → R is positive, i.e., it vanishes only at the origin 0 n . In such a case

B f := {x ∈ R n : f (x) ≤ 1} is a compact convex set.
Theorem 2.1. Let f : R n → R be a positive sublinear function and p ∈ {0, 1, . . . , n}. Then

F → Ψ(F ) = R + (F × {1})
is a bijection between the set of p -dimensional faces of B f and the set of (p + 1) -dimensional faces of epif .

Proof. Let F be a face of B f . We claim that Ψ(F ) is a face of epif . Pick (x, t) and (y, s) in epif and

λ ∈]0, 1[ such that (1 -λ)(x, t) + λ(y, s) ∈ Ψ(F ). (2)
One may suppose that t ̸ = 0, otherwise (x, t) = 0 n+1 and we are done. Similarly, one may suppose that s ̸ = 0. The condition (2) says that

(1 -λ)x + λy = αz (1 -λ)t + λs = α
with α ≥ 0 and z ∈ F . Hence, α ̸ = 0 and

z = [ (1 -λ)t (1 -λ)t + λs ] x t + [ λs (1 -λ)t + λs ] y s
is a convex combination of two vectors in B f . Since F is a face of B f , it follows that x/t and y/s are in F . This proves that (x, t) and (y, s) are in Ψ(F ) and completes the proof of our claim.

Conversely, let G be a nonzero face of epif . There exists a compact convex set F in R n such that Ψ(F ) = G. Such F is unique and given by

F = {x ∈ R n : (x, 1) ∈ G}. ( 3
) We claim that (3) is a face of B f . Pick x, y ∈ B f and λ ∈]0, 1[ such that (1 -λ)x + λy ∈ F . Hence, (1 -λ)(x, 1) + λ(y, 1) ∈ G.
Since G is a face of epif , it follows that (x, 1) and (y, 1) are in G. Therefore, x and y are in F . This proves that F is a face of B f . For completing the proof of the theorem we check that

dim[Ψ(F )] = dim(F ) + 1. (4)
The relation ( 4) is probably known since it holds for any nonempty compact convex set F in R n , and not just for a face of B f . Let p be the dimension of F . Then,

F ⊂ u 0 + {u 1 , . . . , u p }
for suitable vectors {u k } p k=0 in R n . Hence, any element of Ψ(F ) can be expressed in the form

(x, t) = t ( u 0 + p ∑ k=1 λ k u k , 1 ) = t (u 0 , 1) + p ∑ k=1 tλ k (u k , 0)
with t ≥ 0 and λ 1 , . . . , λ p ∈ R. Thus, Ψ(F ) ⊂ span{(u 0 , 1) , (u 1 , 0) , . . . , (u p , 0)}.

This shows that the dimension of Ψ(F ) is at most p + 1. On the other hand, since dim(F ) = p, it is possible to find vectors {v 0 , v 1 , . . . , v p } in F such that {(v 0 , 1), (v 1 , 1), . . . , (v p , 1)} are linearly independent. Hence, Ψ(F ) contains p + 1 linearly independent vectors, and therefore its dimension is at least p + 1. This completes the proof of (4).

Remark . The case p = 0 is of special interest because it concerns the identification of the extreme rays of an epigraphical cone. This theme has been treated already in [21, Corollary 2.2]. The case p = n yields the well known formula

epif = R + (B f × {1}) (5)
for the epigraph of a positive sublinear function.

By combining Theorem 2.1 and the next lemma one can identify the faces of a proper cone, not necessarily an epigraphical one. As many authors do, we say that a closed convex cone is proper if it solid and pointed. Lemma 2.2. For a closed convex cone K in R n+1 the following statements are equivalent:

(a) K is proper.

(b) There exists v ∈ int(K) such that ⟨v, z⟩ > 0 for all nonzero z ∈ K.

(c) K is orthogonally equivalent to the epigraph of a positive sublinear function f : R n → R.

Proof. (a) ⇒ (b). Let K + denote the dual cone of K. Since K ∩ -K + = {0 n+1 } , ( 6
)
the convex cone P = K + K + is closed. We claim that P is pointed. To see this, take a ∈ P ∩ -P and write

a = b 1 + c 1 = -(b 2 + c 2 ) with b 1 , b 2 ∈ K and c 1 , c 2 ∈ K + . Hence, K ∋ (b 1 + b 2 ) = -(c 1 + c 2 ) ∈ -K + .
In view of (6), one gets b 1 = -b 2 and c 1 = -c 2 . Since K and K + are pointed, it follows that b 1 = b 2 = c 1 = c 2 = 0 n+1 . This yields a = 0 n+1 and proves the pointedness of P . As a consequence,

P + = (K + K + ) + = K + ∩ K
is solid. To see that (b) holds, one just needs to pick any v from the interior of P + . (b) ⇒ (c). Let v ∈ int(K) be such that ⟨v, z⟩ > 0 for all z ∈ K\{0 n+1 }. [START_REF] Faraut | Analysis on Symmetric Cones[END_REF] Without loss of generality one may assume that ∥v∥ = 1. Let {u 1 , . . . , u n } be an orthonormal basis for the linear subspace

v ⊥ = {z ∈ R n+1 : ⟨v, z⟩ = 0}. Then U = [u 1 , . . . , u n , v] ∈ O n+1 and U T v = e n+1 := (0 n , 1)
belongs to the interior of Q = U T (K). In view of [START_REF] Faraut | Analysis on Symmetric Cones[END_REF], the closed convex cone Q is contained in the half-space R n × R + . Hence, Q is the epigraph of the nonnegative sublinear function

x ∈ R n → f Q (x) = min{t ∈ R : (x, t) ∈ Q}.
It remains to check that f Q is positive. Suppose, on the contrary, that f Q vanishes at some nonzero vector x. Hence, the linear combination

z = ∑ n i=1 x i u i is a nonzero vector in K such that ⟨v, z⟩ = n ∑ i=1 x i ⟨v, u i ⟩ = 0, contradicting (7)

Rotundity and smoothness

Rotundity and smoothness are fundamental properties concerning the unit ball of a normed space. Such notions can be extended to general convex bodies and even to proper cones. The next definition can be found for instance in [START_REF] Stern | Invariant ellipsoidal cones[END_REF].

Definition 3.1. A proper cone K in an Euclidean space is rotund if every face of K, other than K itself and {0}, is a half-line (called an extreme ray).

Rotund cones are often times referred to as strictly convex cones because they are characterized by the strict convexity condition

z, v ∈ K not collinear, λ ∈]0, 1[ =⇒ (1 -λ)z + λv ∈ int(K).
Rotund cones play an important role in mathematical economics [START_REF] Radner | Paths of economic growth that are optimal with regard only to final states: a turnpike theorem[END_REF] and other fields [START_REF] Gritzmann | Cross-positive matrices revisited[END_REF][START_REF] Schneider | On the invariant faces associated with a cone-preserving map[END_REF]. A nice example of rotund cone is the elliptic cone

E(Q) := {(x, t) ∈ R n+1 : √ ⟨x, Qx⟩ ≤ t}
associated to a positive definite symmetric matrix Q of order n. Also the ℓ p -cone

K p = {(x, t) ∈ R n+1 : ∥x∥ p ≤ t} is rotund if one takes p ∈]1, ∞[
. By contrast, the choices p = 1 or p = ∞ lead to proper cones that are not rotund. All this can be explained in a clear-cut manner with the help of the next proposition.

Proposition 3.2. For a positive sublinear function f : R n → R the following statements are equivalent:

(a) epif is rotund.

(b) f is rotund in the sense that f (x) = 1, f (y) = 1, x ̸ = y imply f (x + y) < 2.
(c) The boundary of B f contains no line-segment.

Proof. (a) ⇔ (c) is a consequence of Theorem 2.1 and (c) ⇔ (b) is obvious.

Example 3.3. The intersection of two elliptic cones is a rotund epigraphical cone. Indeed, if Q 1 and Q 2 are positive definite symmetric matrices of order n, then the intersection

E(Q 1 ) ∩ E(Q 2 ) is
an epigraphical cone whose associated nonnegative sublinear function

x ∈ R n → f (x) = max { √ ⟨x, Q 1 x⟩, √ ⟨x, Q 2 x⟩
} is positive and rotund.

The next corollary is helpful when it comes to check the rotundity of a proper cone that is not necessarily an epigraphical cone.

Corollary 3.4. A proper cone K in R n+1 is rotund if and only if it is orthogonally equivalent to the epigraph of a rotund positive sublinear function

f : R n → R.
Proof. It is a matter of combining Lemma 2.2 and Proposition 3.2. A key observation is that the notion of rotundity for proper cones is invariant under orthogonal transformations.

Remark . The notion of local uniform rotundity is one of the main topics in renorming theory [START_REF] Deville | Smoothness and Renormings in Banach Spaces[END_REF][START_REF] Moltó | A Nonlinear Transfer Technique for Renorming[END_REF]. It has been traditionally considered only as a matter for norms. However, the definition can be easily generalized for positive sublinear functions. This opens the way to the definition of local uniform rotundity for proper cones. For instance, one may declare a proper cone K in R n+1 to be locally uniformly rotund if it is orthogonally equivalent to the epigraph of a locally uniformly rotund positive sublinear function f : R n → R.

There is no universally accepted definition of smoothness for proper cones. The notion of smoothness that we adopt here is not that of [START_REF] Kuriki | Weights of chi-square distribution for smooth or piecewise smooth cone alternatives[END_REF], but one that is dual to rotundity.

Definition 3.5. A proper cone K in an Euclidean

space is smooth if its dual cone K + is rotund.
The most bothering aspect of Definition 3.5 is the need of knowing K + , a cone that is not always directly available. Anyhow, the next proposition clarifies the geometric meaning of smoothness. As one can see, it is possible to check whether an epigraphical cone is smooth without evaluating its dual cone.

x ∈ R n → f (x) = (f • ) • (x) = max y̸ =0n ⟨y, x⟩ f • (y) is then differentiable on R n \{0 n }.
In fact, by using Danskin's differentiability theorem [START_REF] Danskin | The theory of max-min, with applications[END_REF] one sees that the gradient of f at a given point x ∈ R n \{0 n } is the unique solution to the maximization problem

f (x) = max y∈B f • ⟨y, x⟩.
(b) ⇒ (c). This implication is easy and well known. (c) ⇒ (a). The theory of convex bodies asserts that if each boundary point of B f has a unique supporting hyperplane, then the boundary of the polar set

(B f ) • = {y ∈ R n : ⟨y, x⟩ ≤ 1 for all x ∈ B f } contains no line-segment. Since (B f ) • = B f • , it follows that f • is rotund. A posteriori, also f 3 is rotund.
By combining (8) and Proposition 3.2, one sees that epif is smooth.

Example 3.7. The Minkowski sum of two elliptic cones is a smooth epigraphical cone. Indeed, if Q 1 and Q 2 are positive definite symmetric matrices of order n, then E(Q 1 )+E(Q 2 ) is an epigraphical cone whose associated nonnegative sublinear function

x ∈ R n → f (x) = min u+v=x { √ ⟨u, Q 1 u⟩ + √ ⟨v, Q 2 v⟩
} is positive and smooth. Also the vertical sum

E(Q 1 ) ⊕ v E(Q 2 ) = { (x, t) ∈ R n+1 : √ ⟨x, Q 1 x⟩ + √ ⟨x, Q 2 x⟩ ≤ t } is a smooth epigraphical cone, but the intersection E(Q 1 ) ∩ E(Q 2 )

Approximation results

An epigraphical cone is not necessarily pointed, but it can always be approximated in the Painlevé-Kuratowski sense by a sequence of pointed epigraphical cones. This is the idea behind the formulation of the next lemma. In the sequel, the notation lim k→∞ C k stands for the Painlevé-Kuratowski limit of a sequence {C k } k∈N of nonempty sets in an Euclidean space. The definition and main properties of Painlevé-Kuratowski limits can be consulted in standard books on set convergence [START_REF] Aubin | Set-Valued Analysis[END_REF][START_REF] Rockafellar | Variational Analysis[END_REF]. Since we are working in a finite dimensional setting, convergence in the Painlevé-Kuratowski sense is equivalent to convergence with respect to the uniform metric

ϱ(K 1 , K 2 ) := max ∥z∥=1 |dist[z, K 1 ] -dist[z, K 2 ]| ,
or with respect to any other equivalent metric for that matter (cf. [START_REF] Iusem | Distances between closed convex cones: old and new results[END_REF]).

Lemma 4.1. Let f : R n → R be a nonnegative sublinear function. Then

epif = lim k→∞ epif k = cl [ ∞ ∪ k=1 epif k ] ( 9
)
where f k : R n → R is the positive sublinear function given by f k (x) = max{f (x), k -1 ∥x∥}.

Proof. That f k is a positive sublinear function is clear. The sequence {f k } k≥1 is pointwisely nonincreasing and

f (x) = lim k→∞ f k (x) = inf k≥1 f k (x)
for all x ∈ R n . This implies the Painlevé-Kuratowski convergence of the sequence {epif k } k∈N and the formulas stated in [START_REF] Güler | Barrier functions in interior point methods[END_REF].

The geometric interpretation of ( 9) is as follows: what we are doing is to approximate epif by another epigraphical cone that is smaller, namely epif k = (epif ) ∩ R k . Here

R k := {(x, t) ∈ R n+1 : ∥x∥ ≤ kt}
is a revolution cone whose axis is generated by e n+1 . The half-aperture angle of R k is equal to

θ k = arccos ( 1/ √ 1 + k 2 ) ,
i.e., a positive number smaller than π/2. Note that epif k is pointed because R k is pointed. The next approximation result can be found in [START_REF] Seeger | On eigenvalues induced by a cone constraint[END_REF]. We give here a shorter proof that is based on the theory of epigraphical cones. Theorem 4.2. Let K be a closed convex cone in an Euclidean space X. Suppose that K is not a sublinear space. Then (a) There exists an upward monotonic sequence {Q k } k∈N of pointed closed convex cones in X such that K = lim k→∞ Q k .

(b) There exists a downward monotonic sequence {P k } k∈N of solid closed convex cones in X such that K = lim k→∞ P k .

Proof. Since X can be taken as the linear space spanned by K, there is no loss of generality in assuming that K is solid. To avoid trivialities we assume also that dimX ≥ 2. If one sets n = dimX -1, then one can identify X with R n+1 . In view of [10, Proposition 2.8], there exist U ∈ O n+1 and a nonnegative sublinear function f : R n → R such that K = U (epif ). Hence, one can approximate K by means of the pointed closed convex cone

Q k = U (epif k ) with f k as in Lemma 4.1. Note that Q k ⊂ Q k+1 for all k ∈ N, and
lim k→∞ Q k = U ( lim k→∞ epif k ) = U (epif ) = K.
Consider now the part (b). Since K + is not a linear subspace either, there exists an upward monotonic sequence {W k } k∈N of pointed closed convex cones in X such that

K + = lim k→∞ W k . If one sets P k = W + k , then {P k } k∈N is a downward monotonic sequence of solid closed convex cones in X such that lim k→∞ P k = ( lim k→∞ W k ) + = (K + ) + = K,
the leftmost equality being a consequence of the Walkup-Wets isometry theorem [START_REF] Walkup | Continuity of some convex-cone-valued mappings[END_REF].

Another approximation result in the same spirit concerns the possibility of approximating a closed convex cone by a rotund cone or by a smooth cone. Theorem 4.3. Let K be a closed convex cone in an Euclidean space X. Suppose that K is not a sublinear space. Then (a) K = lim k→∞ R k for some upward monotonic sequence {R k } k∈N of rotund cones in X.

(b) K = lim k→∞ S k for some downward monotonic sequence {S k } k∈N of smooth cones in X.

Proof. Without loss of generality one can assume that K is solid and that dimX ≥ 2. As before, we set n = dimX -1 and identify X with R n+1 . Then we write

K = U (epif ) R k = U (epi g k ),
where U and f are as in the proof of Theorem 4.2, and

g k (x) = f (x) + k -1 ∥x∥. Note that R k ⊂ R k+1 for all k ∈ N, and lim k→∞ R k = U ( lim k→∞ epi g k ) = U (epif ) = K.
Each R k is pointed because the sublinear function g k is positive. Since K is solid, R k is solid (hence, proper) for all k large enough (cf. [START_REF] Iusem | Pointedness, connectedness, and convergence results in the space of closed convex cones[END_REF]Corollary 5.2]). It remains to check that R k is rotund, but this is a consequence of Proposition 3.2 and the fact that g k is rotund. The part (b) is obtained from (a) by relying on duality arguments.

Theorem 4.4 says that any closed convex cone can be approximated in the Painlevé-Kuratowski sense by a polyhedral cone. Theorem 4.4. Any closed convex cone in an Euclidean space X can be written as Painlevé-Kuratowski limit of a sequence of polyhedral cones in X.

Proof. Let K be a closed convex cone in X. We suppose that K is not a linear subspace, otherwise we are done. Since one can take X as the linear space spanned by K, there is no loss of generality in assuming that K is solid. As in Theorem 4.2, one assumes that dimX ≥ 2, one sets n = dimX -1, and one identifies X with R n+1 . In view of Theorem 4.2, for all k ∈ N there exists a pointed closed convex cone

Q k in R n+1 such that ϱ(Q k , K) ≤ 1/k. We represent Q k = U k (epif k ) in
epif k = R + (B f k × {1}) Note that B f k is a nonempty compact convex set in R n . Thanks to [17, Theorem 2.1], there exists a polytope Ω k in R n such that haus(Ω k , B f k ) ≤ 1/k
with "haus" standing for the classical Pompeiu-Hausdorff metric. Since 0 n ∈ int(B f k ), one may suppose that 0 n ∈ int(Ω k ). Consider now the polyhedral cone

P k = R + (Ω k × {1}) .
Thanks to [12, Proposition 6.3], one has the estimate

ϱ(P k , epif k ) ≤ 2 haus(Ω k , B f k ).
Observe that U k (P k ) is a polyhedral cone and

ϱ(U k (P k ), K) ≤ ϱ(U k (P k ), Q k ) + ϱ(Q k , K) = ϱ (U k (P k ), U k (epif k )) + ϱ(Q k , K) = ϱ(P k , epif k ) + ϱ(Q k , K) ≤ 2(1/k) + (1/k).
This shows that K = lim k→∞ U k (P k ) and completes the proof of the theorem.

Characteristic function of an epigraphical cone

If K is a solid closed convex cone in R d , then its characteristic function Φ K : int(K) → R is defined by the d-dimensional integral Φ K (z) := ∫ K + e -⟨z,v⟩ dv.
Such a definition of characteristic function is discussed in the book [7, Chapter 1] and in many other places [START_REF] Faybusovich | Self-concordant barriers for cones generated by Chebyshev systems[END_REF][START_REF] Güler | Barrier functions in interior point methods[END_REF][START_REF] Vinberg | The theory of homogeneous convex cones[END_REF]. Some authors refer to Φ K as the Vinberg (or Koszul-Vinberg) characteristic function of K.

It is clear that Φ K is positively homogeneous of degree -d. As shown in [7, Proposition I.3.2], a fundamental property of Φ K is that of behaving as barrier function for the cone K. This means that lim

k→∞ Φ K (z k ) = ∞ (10)
for any sequence {z k } k∈N in int(K) converging to a point on the boundary of K. As can be seen from the next lemma, there is also a converse statement: the condition (10) forces {z k } k∈N to approach the boundary of K.

Lemma 5.1. Let K is a solid closed convex cone in R d . Then for all z ∈ int(K), one has

Φ K (z) ≤ ( 1 dist[z, bd(K)] ) d ∫ K + e -∥v∥ dv.
Proof. The point z is at positive distance from bd(K). If r is such distance, then z + rB d ⊂ K. This inclusion amounts to saying that r ∥v∥ ≤ ⟨z, v⟩ for all v ∈ K + . Hence

Φ K (z) ≤ ∫ K + e -r∥v∥ dv.
By using a positive homogeneity argument and a suitable change of variables, the term r moves out of the integral as a factor r -d .

We now address the question of computing the characteristic function of an epigraphical cone. In the sequel one uses the notation

Ξ f • (x, s) := ∫ {f • ≤s} e ⟨x,y⟩ dy, ( 11
)
where integration is carried out over the sublevel set {f

• ≤ s} = {y ∈ R n : f • (y) ≤ s}.
Theorem 5.2. If f : R n → R is a positive sublinear function, then

Φ epif (x, t) = 1 t ∫ R n e ⟨x,y⟩-tf • (y) dy (12) = ∫ ∞ 0 e -ts Ξ f • (x, s)ds ( 13
)
for all (x, t) ∈ int(epif ).

Proof. Let (x, t) be in the interior of E f . Then

Φ epif (x, t) = ∫ epif 3 e -(⟨x,w⟩+ts) dwds (14) = ∫ R n [ ∫ ∞ f 3 (w) e -⟨x,w⟩-ts ds ] dw (15) = ∫ R n [ e -⟨x,w⟩ e -tf 3 (w) t ] dw,
where ( 14) is a consequence of (8), and ( 15) is due to Fubini's integration theorem. The change of variables y = -w leads finally to the formula [START_REF] Iusem | Distances between closed convex cones: old and new results[END_REF]. In order to obtain (13) one integrates [START_REF] Kuriki | Weights of chi-square distribution for smooth or piecewise smooth cone alternatives[END_REF] in a different order. One has

Φ epif (x, t) = ∫ ∞ 0 [ ∫ {f 3 ≤s} e -(⟨x,w⟩+ts) dw ] ds = ∫ ∞ 0 e -ts [ ∫ {f 3 ≤s} e -⟨x,w⟩ dw ] ds.
It suffices now to observe that the inner integral is equal to Ξ f • (x, s).

As one can see from ( 13), the function Φ epif (x, • ) is nothing but the standard Laplace transform of Ξ f • (x, • ). It is not always easy to compute the multidimensional integral [START_REF] Iusem | Pointedness, connectedness, and convergence results in the space of closed convex cones[END_REF], not to mention its Laplace transform, but there are some particular cases were this can be done explicitly. The next example is inspired from [9, Section 7.4].

Example 5.3. Consider the positive sublinear function

f (x) = ∥x∥ 1 , in which case f • (y) = ∥y∥ ∞ . One start by computing Ξ f • (x, s) = ∫ ∥y∥∞≤s e ⟨x,y⟩ dy = n ∏ i=1 ∫ s -s e -x i y i dy i = n ∏ i=1 σ(x i , s).
Here

σ(τ, s) = { (e τ s -e -τ s ) /τ if τ ̸ = 0 2s if τ = 0.
Then one needs to evaluate the Laplace transform at t of the function ∏ n i=1 σ(x i , •). Of course, one supposes that ∥x∥ 1 < t. For each i ∈ {1, . . . , n}, one has to distinguish between the cases x i ̸ = 0 and x i = 0. Consider for instance the configuration x 1 > 0, . . . , x n > 0. Since,

n ∏ i=1 ( e x i s -e -x i s ) = ∑ ε i =±1 ( n ∏ i=1 ε i ) e ∑ n i=1 sε i x i , one ends up with Φ epif (x, t) = [ n ∏ i=1 x i ] -1 ∑ ε i =±1 ∏ n i=1 ε i t - ∑ n i=1 ε i x i .

Moment-generating function techniques

Since f • is positively homogeneous, a simple change of variables in the integral (12) leads to

Φ epif (x, t) = 1 t n+1 ∫ R n e ⟨t -1 x, y⟩-f • (y) dy , which in turn implies that Φ epif (x, t) = 1 t n+1 Φ epif ( x t , 1 ) . ( 16
)
In other words, one can always restrict the attention to the particular case t = 1. By the way, the relation ( 16) is consistent with the fact that Φ epif is positively homogeneous of degree -(n + 1).

Example 5.4. Let f (x) = ∥x∥ ∞ , in which case f • (y) = ∥y∥ 1 . A matter of computation shows that Φ epif (x, 1) = ∫ R n e ⟨x, y⟩ e -∥y∥ 1 dy = n ∏ i=1 ∫ ∞ -∞ e x i τ -|τ | dτ = n ∏ i=1 2 1 -x 2 i whenever ∥x∥ ∞ < 1.
In view of ( 16), one obtains

Φ epif (x, 1) = 2 n t n-1 ∏ n i=1 (t 2 -x 2 i )
whenever ∥x∥ ∞ < t, recovering in this way a formula stated in [9, Lemma 7.3].

It is worthwhile to mention that

x ∈ int(B f ) → M f (x) := Φ epif (x, 1) = ∫ R n
e ⟨x, y⟩ e -f • (y) dy can be seen as the moment-generating function of an n -dimensional random vector distributed according to the density function y → e -f • (y) . Strictly speaking, e -f • is a density function only up to a positive normalization factor. A density functions of the form ce -g , for some constant c and some norm g, generalizes the classical Laplace density function. This way of perceiving M f leads to a number of analytic results that are well known in probability theory. The next corollary is mentioned just by way of illustration. (b) If f is even, then so is M f .

Remark . Partial differentiation of Φ epif with respect to x leads to

∇ x Φ epif (x, t) = 1 t ∫ R n
y e ⟨x,y⟩-tf • (y) dy for all (x, t) ∈ int(epif ). In particular,

∇M f (x) = ∫ R n y e ⟨x,y⟩-f • (y) dy for all x ∈ int(B f ).
The above gradient has a special meaning if one sets x = 0 n . Indeed, the term

∇M f (0 n ) = ∫ R n y e -f • (y) dy
corresponds to a mathematical expectation.

The next proposition provides a formula for computing the characteristic function of an elliptic cone. The moment-generating function

x ∈ int (B n ) → M(x) = ∫ R n
e ⟨x, y⟩ e -∥y∥ dy associated to e -∥•∥ is something intrinsic to the Euclidean space R n and can be computed once and for all. One has

M(x) = κ n [1 -∥x∥ 2 ] (n+1)/2 ,
where the constant κ n is given by

κ n = M(0 n ) = ∫ R n e -∥y∥ dy = π n 2 n! ( n 2 ) Γ ( n 2 )
with Γ standing for the Euler gamma function.

Proposition 5.6. Let Q be a positive definite symmetric matrix of order n. Let {u 1 , . . . , u n } be an orthonormal basis of eigenvectors of Q and let {λ 1 , . . . , λ n } be the corresponding eigenvalues.

Then, for all (x, t) ∈ int[E(Q)], one has

Φ E(Q) (x, t) = 1 t n+1 √ det Q M ( L T x t ) (17) = κ n √ det Q [t 2 -∥L T x∥ 2 ] (n+1)/2 (18)
with L standing for the matrix of order n whose j-th column is the vector √ λ j u j .

Proof. The polar of f (x) = √ ⟨x, Qx⟩ is given by f

• (y) = √ ⟨y, Q -1 y⟩. Hence, Φ E(Q) (x, 1) = ∫ R n e ⟨x,y⟩ e - √ ⟨y, Q -1 y⟩ dy.
By using the spectral decomposition

Q = U DU T = n ∑ j=1 λ j u j u T j
and the orthogonal transformation η = U T y, one gets

Φ E(Q) (x, 1) = ∫ R n e ⟨U T x, η⟩ e - √ ⟨η, D -1 η⟩ dη.
Finally, the change of variables

ξ j = η j / √ λ j leads to Φ E(Q) (x, 1) = √ λ 1 • • • λ n ∫ R n e ⟨L T x
, ξ⟩ e -∥ξ∥ dξ.

This completes the proof of the case t = 1. The formula ( 17) is then obtained by using the homogenization mechanism [START_REF] Moussaoui | Subdifferentiability and inf-sup theorems[END_REF].

Example 5.7. By taking Q = I n as the identity matrix of order n one recovers the well known formula (cf. [START_REF] Faraut | Analysis on Symmetric Cones[END_REF][START_REF] Güler | Barrier functions in interior point methods[END_REF])

Φ Λ (x, t) = κ n [t 2 -∥x∥ 2 ] (n+1)/2
for the characteristic function of the (n + 1) -dimensional Lorentz cone

Λ := {(x, t) ∈ R n+1 : ∥x∥ ≤ t}.
More generally, the choice Q = (tan ϑ) -2 I n leads to the expression

Φ rev(ϑ) (x, t) = κ n tan ϑ [(t tan ϑ) 2 -∥x∥ 2 ] (n+1)/2
for the characteristic function of a revolution cone rev(ϑ) := {(x, t) ∈ R n+1 : (tan ϑ) -1 ∥x∥ ≤ t} with half-aperture angle equal to ϑ.

Epigraphical conic programming

An epigraphical conic program (ECP) is an optimization problem of the form minimize ⟨c, z⟩ with respect to the Karush-Kuhn-Tucker multipliers λ 1 , . . . , λ N ∈ (epif ) + . If one keeps [START_REF] Gritzmann | Cross-positive matrices revisited[END_REF] in mind, then a matter of simplification shows that the dual problem can be written in the form maximize

N ∑ k=1 ⟨b k , λ k ⟩ (20) λ 1 , . . . , λ N ∈ epif 3 N ∑ k=1 A T k λ k = c.
Let v primal and v dual denote the optimal values of ( 19) and [START_REF] Schneider | On the invariant faces associated with a cone-preserving map[END_REF], respectively. The general duality theory for convex optimization problems leads to the following result. ii) f is polyhedral and ( 19) is feasible.

Proof. Under the assumption (ii), one can convert [START_REF] Rockafellar | Variational Analysis[END_REF] and ( 20) into a pair of dual linear programs, one of which is feasible. Hence, the other problem is feasible as well, and both problems have the same optimal value. Under the Slater type qualification hypothesis (i), the dual problem (20) admits a solution and both problems have the same optimal value. This can be shown by using one of the many existing min-max theorems for convex-concave Lagrangeans, see for instance [16, Corollary 4.2].

First application: finding a Chebishev center

Consider a finite set Ω = {ω 1 , . . . , ω N } of distinct points in R n . One wishes to find a vector x ∈ R n that is regarded as the "center" of Ω. There are different ways of formalizing the concept of center. The Chebishev strategy consists in minimizing the maximal deviation function

x ∈ R n → δ max (x) := max 1≤k≤N f (x -ω k ) ,
where f is a given norm on R n . By definition, a Chebishev center of Ω is a solution to the minimization problem r cheb = min x∈R n δ max (x). The number r cheb is called the Chebishev radius of Ω. Since δ max is continuous and has bounded level sets, the existence of a Chebishev center is automatically guaranteed. On the other hand, minimizing δ max is clearly equivalent to solving minimize t

(x, t) ∈ R n+1 f (x -ω k ) ≤ t for all k ∈ {1, . . . , N }.
This can be written of course in the conic programming format minimize t (

(x, t) ∈ (ω k , 0) + epif for all k ∈ {1, . . . , N }.

Since ( 21) is a particular instance of the general ECP, one obtains the following conclusion.

Corollary 6.2. Let f be a norm on R n . Then the Chebishev radius of Ω is equal to the optimal value of the maximization problem maximize

N ∑ k=1 ⟨ω k , y k ⟩ (22) (y k , s k ) ∈ epif • for all k ∈ {1, . . . , N } N ∑ k=1 y k = 0 n , N ∑ k=1 s k = 1,
where the decision variables are the vectors y 1 , . . . , y N ∈ R n and the scalars s 1 , . . . , s N .

Proof. For obtaining the dual problem [START_REF] Seeger | On eigenvalues induced by a cone constraint[END_REF], one just needs to work out the general model [START_REF] Schneider | On the invariant faces associated with a cone-preserving map[END_REF] with

z = [ x t ] , c = [ 0 n 1 ] , b k = [ ω k 0 ] , A k = I n+1 .
For the sake of matrix calculus we are writing the vectors of R n+1 in column notation. The KKT multipliers are the (n + 1)-dimensional vectors

λ 1 = [ y 1 s 1 ] , . . . , λ N = [ y N s N ] ,
which we identify with the pairs (y 1 , s 1 ), . . . , (y N , s N ). Note that the Slater type qualification condition (i) mentioned in Lemma 6.1 holds automatically in the present context. Indeed, if one picks any x ∈ R n and t bigger than max 1≤k≤N f (x -ω k ), then

A k z -b k = [ x t ] - [ ω k 0 ] ∈ int (epif )
for all k ∈ {1, . . . , N }.

The maximization problem ( 22) is perhaps better understood if one writes it in the form maximize

N ∑ k=1 s k ⟨ω k , u k ⟩ s k ≥ 0, f • (u k ) ≤ 1 for all k ∈ {1, . . . , N } N ∑ k=1 s k u k = 0 n , N ∑ k=1 s k = 1,
where the maximum is taken over all the representations of 0 n as convex combination of N vectors in the ball B f • associated to the polar norm f • .

Second application: finding a Fermat center

Instead of the maximal deviation function δ max one could use the total deviation function

x ∈ R n → δ sum (x) := N ∑ k=1 f (x -ω k ).
The existence of a minimum of δ sum is not problematic either. By definition, a Fermat center of Ω is a solution to the minimization problem

r fermat = min x∈R n δ sum (x). ( 23
)
The above minimal value is called the Fermat radius of Ω. Following [START_REF] Boyd | Applications of second-order cone programming[END_REF], we write the unconstrained problem [START_REF] Stern | Invariant ellipsoidal cones[END_REF] Proof. We apply again Lemma 6.1, but this time we use

z = [ x τ ] , c = [ 0 n 1 N ] , b k = [ ω k 0 ] , A k = [ I n O n,N 0 T n e T k,N
] .

Here τ is the column vector whose components are τ 1 , . . . , τ N , the symbol O n,N indicates the zero matrix of size n × N , and e k,N is the k-th canonical vector of R N . The equality constraint in [START_REF] Schneider | On the invariant faces associated with a cone-preserving map[END_REF] becomes

N ∑ k=1 [ I n 0 n O N,n e k,N ] [ y k s k ] = [ 0 n 1 N ] ,
which after simplification yields ∑ N k=1 y k = 0 n and s k = 1 for all k ∈ {1, . . . , N }. This explains the form (25) that we are getting for the dual problem associated to [START_REF] Vinberg | The theory of homogeneous convex cones[END_REF]. The Slater type qualification condition (i) mentioned in Lemma 6.1 is again in force. To see this, pick any x ∈ R n and then let τ1 > f (x -ω 1 ), . . . , τN > f (x -ω N ). In such a case,

A k z -b k = [ x τk
] -

[ ω k 0 ] ∈ int (epif )
for all k ∈ {1, . . . , N }.

Final comments

The nice survey by Boyd et al. This is of course a particular instance of the general model [START_REF] Rockafellar | Variational Analysis[END_REF].

  . Summarizing, we have shown that K is orthogonally equivalent to the epigraph of the positive sublinear function f Q . (c) ⇒ (a). It follows from [21, Proposition 2.1] and the fact that properness is invariant under orthogonal transformations.

Proposition 3 . 6 .

 36 For a positive sublinear function f : R n → R the following statements are equivalent: (a) epif is smooth. (b) f is smooth, i.e., differentiable on R n \{0 n }.

  (c) Every boundary point of B f admits exactly one supporting hyperplane. Proof. (a) ⇒ (b). By assumption, the dual cone of epif is rotund. But (epif ) + = epif 3 (8) with f 3 : R n → R denoting the skewed polar of f (cf. [21, Section 2.3]). Recall that f 3 (y) = f • (-y), where f • is the usual polar function. By Proposition 3.2, the positive sublinear function f 3 is rotund. Hence, f • is rotund. By a classical duality argument,

Corollary 5 . 5 .

 55 Let f : R n → R be a positive sublinear function. Then (a) M f is strictly logconvex and infinitely often differentiable on int(B f ).

A

  k z -b k ∈ epif for all k ∈ {1, . . . , N } with f : R n → R standing for a positive sublinear function. The decision variable z is a vector in some Euclidean space, say R d . The matrices A 1 , . . . , A N and the vectors c, b 1 , . . . , b N are known and have appropriate dimensions. A dual problem to (19) can be constructed by using the standard Lagrangean formalism. If one introduces a Lagrangean function L of the form L(z, λ 1 , . . . , λ N ) = ⟨c, z⟩ + N ∑ k=1 ⟨λ k , b k -A k z⟩, then the dual problem to (19) consists in maximizing the term inf z∈R d L(z, λ 1 , . . . , λ N )

Lemma 6 . 1 .

 61 Let f : R n → R be a positive sublinear function. Then the equality v primal = v dual holds under any of the following two qualification hypotheses: i) There exists z ∈ R d such that A k z -b k ∈ int (epif ) for all k ∈ {1, . . . , N }.

fCorollary 6 . 3 .

 63 (x -ω k ) ≤ τ k for all k ∈ {1, . . . , N }, where the decision variables are the components of x ∈ R n and the auxiliary variables τ 1 , . . . , τ N . The latter problem can be reformulated as minimize τ k ) ∈ (ω k , 0) + epif for all k ∈ {1, . . . , N }, so we are dealing with yet another particular case of the general ECP. Let f be a norm on R n . Then the Fermat radius of Ω is equal to the optimal value of the maximization problem maximizeN ∑ k=1 ⟨ω k , y k ⟩ (25) f • (y k ) ≤ 1 for all k ∈ {1, . . . , N } N ∑ k=1 y k = 0 n .

[ 3 ]

 3 on second-order conic programming (SOCP) focuses the attention on the optimization model minimize ⟨w, x⟩ (26) ∥A k x + b k ∥ ≤ ⟨c k , x⟩ + d k for all k ∈ {1, . . . , N }.

  Let K be a nontrivial closed convex cone in R n+1 . Then K is solid if and only if there exist U ∈ O n+1 and a nonnegative sublinear function f : R n → R such that

Definition 1.1. Let O d denote the group of orthogonal matrices of order d. Two convex cones K 1 , K 2 in the Euclidean space R d are orthogonally equivalent if there exists U ∈ O d such that K 2 = U (K 1 ). Lemma 1.2.

  Proof. Combine Lemma 2.2, Proposition 3.6, and the fact that the notion of smoothness for proper cones is invariant under orthogonal transformations.

may not be smooth. The next corollary fully characterizes the class of smooth cones. Corollary 3.8. A proper cone K in R n+1 is smooth if and only if it is orthogonally equivalent to the epigraph of a smooth positive sublinear function f : R n → R.

  terms of a suitable matrix U k ∈ O n+1 and a corresponding positive sublinear function f k : R n → R. As in[START_REF] Deville | Smoothness and Renormings in Banach Spaces[END_REF], one has

SOCP are typically solved by interior point methods (cf. [2,3,13]). Such methods can be adapted to the epigraphic conic program (19), provided one has a suitable barrier function for the set epif . It is here where the Vinberg characteristic function Φ epif enters into action. It would lead us to far to describe all the technicalities of the interior point method applied to (19), but this is certainly something that deserves an exhaustive treatment in a more specialized paper.


Alberto Seeger
email: alberto.seeger@univ-avignon.fr

Epigraphical cones I

Keywords: Mathematics Subject Classification: 46B10, 46B20, 52A41 Convex cone, epigraphical cone, sublinear function, inradius of a cone, solidity, pointedness, angular spectrum

Introduction

The purpose of this work is to survey the class of epigraphical cones and to derive new results concerning the geometric structure of these mathematical objects. A convex cone in the Euclidean space R n+1 is an epigraphical cone if it can be represented as epigraph epif = {(x, t) ∈ R n+1 : f (x) ≤ t} of a nonnegative sublinear function f : R n → R. Sublinearity is understood as combination of subadditivity and positive homogeneity. If K is an epigraphical cone in R n+1 , then its associated nonnegative sublinear function is given by

f K (x) = min{t ∈ R : (x, t) ∈ K}.
An epigraphical cone is necessarily closed and nontrivial, i.e., different from the whole space and different from the zero cone. It is also solid in the sense that it has a nonempty topological interior. Conversely, up to orthogonal transformation, every nontrivial solid closed convex cone is an epigraphical cone (cf. [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF]Proposition 2.8]).

Epigraphical cones are important enough to justify a preferential and exhaustive treatment. A few striking examples are appropriate to illustrate this point. In the sequel ⟨•, •⟩ indicates the usual inner product of R d , regardless of the dimension d.

Example 1.1. The most prominent example of epigraphical cone is

E(Q) := {(x, t) ∈ R n+1 : √ ⟨x, Qx⟩ ≤ t}
with Q standing for a positive definite symmetric matrix of order n. One refers to E(Q) as the elliptic cone associated to Q. Elliptic cones have been studied under different angles in a number of references, including our own works [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 | Angular analysis of two classes of non-polyhedral convex cones: the point of view of optimization theory[END_REF] and those of Stern and Wolkowicz [START_REF] Stern | Exponential nonnegativity on the ice cream cone[END_REF][START_REF] Stern | Invariant ellipsoidal cones[END_REF]. See [START_REF] Bhattacharya | Ellipsoidal cones and rendezvous of multiple agents[END_REF] for an application of elliptic cones in control theory. Elliptic cones can be defined also in Hilbert spaces (cf. [START_REF] Burton | Contracting the maximal points of an ordered convex set[END_REF]), but we stick to finite dimensionality.

Example 1.2. Another interesting example of epigraphical cone is the ℓ p -cone

K p := {(x, t) ∈ R n+1 : ∥x∥ p ≤ t}.
Here p ∈ [1, ∞] and ∥ • ∥ p denotes the ℓ p -norm in R n . Basic information on ℓ p -cones can be found in [START_REF] Barker | Cones of diagonally dominant matrices[END_REF][START_REF] Boyd | Convex Optimization[END_REF][START_REF] Güler | Barrier functions in interior point methods[END_REF][START_REF] Lyubich | Perron-Frobenius theory for finite-dimensional spaces with a hyperbolic cone[END_REF].

Example 1.3. Finally, consider an epigraphical cone that is polyhedral. Such a cone has the particular structure

P C := { (x, t) ∈ R n+1 : max 1≤i≤m ⟨c i , x⟩ ≤ t } ,
where C = {c 1 , . . . , c m } is a finite subset of R n such that 0 n ∈ co(C). The symbol 0 n stands for the zero vector of R n and "co" refers to the convex hull operation. Note that K 1 and K ∞ fit into this category of cones.

The above list of examples gives already a good idea on the kind of convex cones we are interested in. The class of epigraphical cones is wider than the class of top-heavy cones introduced by Fiedler and Haynsworth [START_REF] Fiedler | Cones which are topheavy with respect to a norm[END_REF]. Recall that a top-heavy cone is understood as an epigraphical cone associated to a norm. Our definition of epigraphical cone slightly deviates from that of [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF]. Indeed, for the sake of simplicity in the overall exposition we have decided not to work with extended-real-valued sublinear functions. The use of extended-real-valued sublinear functions would allow to consider, for instance, the class of L p -cones in the sense of Glineur [START_REF] Glineur | Topics in convex optimization: interior-point methods, conic duality and approximations[END_REF][START_REF] Glineur | Conic formulation for l p -norm optimization[END_REF]. The organization of the paper is as follows:

-Section 2 develops the basic algebra of epigraphical cones.

-Section 3 provides some rules for computing the inradius of an epigraphical cone. The inradius is a coefficient that serves to measure the degree of solidity of the cone.

-Section 4 is about measuring the degree of pointedness of an epigraphical cone.

-Section 5 analyzes the angular structure of an epigraphical cone.

Other aspects concerning the theory of epigraphical cones are treated in the companion paper [START_REF] Seeger | Epigraphical cones II[END_REF]. The emphasis in [START_REF] Seeger | Epigraphical cones II[END_REF] is put in the study of properties that are valid up to orthogonal characterizations, which allows to consider a class of convex cones larger than the class of epigraphical cones. The notation that we use in both parts is standard or self-explanatory: int(Ω), bd(Ω), cl(Ω) indicate respectively the interior, boundary, and closure of a set Ω. The unit sphere and the closed unit ball of R d are denoted by S d and B d , respectively.

The basic algebra of epigraphical cones 2.1 Representation of epif in terms of a convex base

Sometimes it is useful to write a sublinear function f : R n → R as support function

x ∈ R n → f (x) = Ψ * C (x) := max y∈C ⟨y, x⟩ of a convex compact set C in R n .
Hörmander's theorem asserts that C is unique and given by

C = {y ∈ R n : ⟨y, x⟩ ≤ f (x) for all x ∈ R n } .
In the parlance of convex analysis, this set corresponds to the subdifferential of f at 0 n . For this reason one writes f = Ψ * ∂f (0n) . If the sublinear function f is nonnegative, then ∂f (0 n ) contains 0 n and admits the polar representation ∂f

(0 n ) = B • f with B f := {x ∈ R n : f (x) ≤ 1} B • := {y ∈ R n : ⟨y, x⟩ ≤ 1 for all x ∈ B}.
As shown in the next proposition, an epigraphical cone in R n+1 is the closed positive hull of a Cartesian product of the form B × {1} with B standing for a closed convex set in R n . Recall that a closed convex cone is pointed if it does not contain a line. Proposition 2.1. Let f : R n → R be a nonnegative sublinear function. Then

epif = cl   ∪ t≥0 t (B f × {1})   . (1)
Furthermore, (a) The "upward" canonical vector e n+1 = (0 n , 1) belongs to int (epif ).

(b) If f vanishes only at 0 n , then the closure operation in ( 1) is superfluous and epif is pointed.

(c) epif is polyhedral if and only if B f is polyhedral (possibly the whole space R n ).

Proof. The representation formula (1) is stated already in [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF]. Such equality yields in particular

int(B f ) × {1} ⊂ int(epif ).
This proves (a) because 0 n ∈ int(B f ). That f vanishes only at 0 n is equivalent to saying that B f is compact, in which case also B f × {1} is compact and the closure operation in (1) is superfluous. The pointedness of epif follows by a simple inspection. The part (c) can be obtained by combining Corollaries 19.2.1 and 19.2.2 of Rockafellar's book [START_REF] Rockafellar | Convex Analysis[END_REF].

Remark . The minimal hypothesis that warrants the pointedness of ( 1) is the sharpness of f . A nonnegative sublinear function

f : R n → R is called sharp if f (x) = 0 and f (-x) = 0 imply x = 0 n .
This property is weaker than saying that f vanishes only at 0 n .

Recall that a generator (or extreme vector) of a pointed closed convex cone K is a nonzero vector e ∈ K such that z ∈ K and e -z ∈ K imply that z ∈ R + e. In such a case one refers to R + e as an extreme ray of K.

Combining two epigraphical cones

Epigraphical cones can be combined in various ways in order to produce new epigraphical cones. For instance, the intersection and the Minkowski sum of two epigraphical cones are again epigraphical cones. More precisely,

epif 1 ∩ epif 2 = epi(f 1 ∨ f 2 ) (2) epif 1 + epif 2 = epi(f 1 f 2 ) (3)
with ∨ and standing, respectively, for the pointwise maximum and the infimal convolution operation. The formula (2) for the intersection is obvious. The formula (3) for the Minkowski sum follows from the infimal convolution theory as developed by Moreau [START_REF] Moreau | Fonctionelles Convexes, Equations aux derivées partielles[END_REF] and Rockafellar [START_REF] Rockafellar | Convex Analysis[END_REF].

Example 2.3. For each x ∈ R n , the function p ∈ [1, ∞] → ∥x∥ p is nonincreasing. One gets ∥ • ∥ p 1 ∨ ∥ • ∥ p 2 = ∥ • ∥ min{p 1 ,p 2 } ∥ • ∥ p 1 ∥ • ∥ p 2 = ∥ • ∥ max{p 1 ,p 2 } for all p 1 , p 2 ∈ [1, ∞]. Hence, K p 1 ∩ K p 2 = K min{p 1 ,p 2 } and K p 1 + K p 2 = K max{p 1 ,p 2 } .
The intersection of two elliptic cones is not necessarily an elliptic cone. However, one gets an epigraphical cone that has a special structure. The same remark applies to the Minkowski sum of two elliptic cones. The next proposition explains the details. In the sequel the symbol

E Q := { y ∈ R n : ⟨y, x⟩ ≤ √ ⟨x, Qx⟩ for all x ∈ R n } (4)
denotes the ellipsoid associated to a positive definite symmetric matrix Q. Clearly one has

E(Q) = epi(Ψ * E Q ).
According to the references [START_REF] Lee | A volume problem for an n-dimensional ellipsoid intersecting with a hyperplane[END_REF][START_REF] Seeger | Direct and inverse addition in convex analysis and applications[END_REF], two alternative characterizations for the ellipsoid (4) are

E Q = Q 1/2 (B n ) = { y ∈ R n : ⟨y, Q -1 y⟩ ≤ 1 } ,
where Q 1/2 denotes the symmetric square root of Q.

Proposition 2.4. Let Q 1 , Q 2 be positive definite symmetric matrices of order n. Then

E(Q 1 ) ∩ E(Q 2 ) = epi(f ) E(Q 1 ) + E(Q 2 ) = epi(g) with f = Ψ * co(E Q 1 ∪E Q 2 ) and g = Ψ * E Q 1 ∩ E Q 2 .
Proof. This result is a direct consequence of the formulas (2)-(3). Note that

Ψ * co(E Q 1 ∪ E Q 2 ) = Ψ * E Q 1 ∨ Ψ * E Q 2 Ψ * E Q 1 ∩ E Q 2 = Ψ * E Q 1 Ψ * E Q 2 and that x ∈ R n → Ψ * E Q k (x) = √ ⟨x, Q k x⟩ corresponds to the nonnegative sublinear function associated to the elliptic cone E(Q k ).
There are also other ways of combining sets in a product space. For instance, if M 1 , M 2 are sets in R n+1 , then one can define their vertical sum and their horizontal sum respectively by

M 1 ⊕ v M 2 := { (x, t) ∈ R n+1 : ∃ r ∈ R s.t. (x, r) ∈ M 1 and (x, t -r) ∈ M 2 } , M 1 ⊕ h M 2 := { (x, t) ∈ R n+1 : ∃ u ∈ R n s.t. (u, t) ∈ M 1 and (x -u, t) ∈ M 2 } .
These operations are mentioned in [START_REF] Rockafellar | Convex Analysis[END_REF]Section 3], though under a different terminology. The next proposition shows that the vertical sum and the horizontal sum of two epigraphical cones are again epigraphical cones. Recall that

C 1 △C 2 := ∪ r∈[0,1] (1 -r)C 1 ∩ rC 2
is the inverse sum of two sets C 1 , C 2 in R n and that

x ∈ R n → (f 1 △f 2 )(x) := min u∈R n max{f 1 (x -u), f 2 (u)}
is the inverse sum of two nonnegative sublinear functions f 1 , f 2 : R n → R. General information about inverse addition can be found in [START_REF] Seeger | Direct and inverse addition in convex analysis and applications[END_REF][START_REF] Seeger | On a convolution operation obtained by adding level sets: classical and new results[END_REF].

Proposition 2.5. Let f 1 , f 2 : R n → R be nonnegative sublinear functions. Then the usual sum

f 1 + f 2 : R n → R and the inverse sum f 1 △f 2 : R n → R are nonnegative sublinear functions. Furthermore, epif 1 ⊕ v epif 2 = epi(f 1 + f 2 ) epif 1 ⊕ h epif 2 = epi(f 1 △f 2 ).
Proof. From the definition of the vertical sum, one sees that

(x, t) ∈ epif 1 ⊕ v epif 2 ⇐⇒ f 1 (x) + f 2 (x) ≤ t ⇐⇒ (x, t) ∈ epi(f 1 + f 2 ).
Similarly, the definition of the horizontal sum shows that

(x, t) ∈ epif 1 ⊕ h epif 2 ⇐⇒ ∃ u ∈ R n s.t. max{f 1 (x -u), f 2 (u)} ≤ t ⇐⇒ (x, t) ∈ epi(f 1 △f 2 ).
It is clear that f 1 + f 2 and f 1 △f 2 are nonnegative sublinear functions. In fact,

f 1 + f 2 = Ψ * ∂f 1 (0n) + Ψ * ∂f 2 (0n) = Ψ * ∂f 1 (0n)+∂f 2 (0n) , f 1 △ f 2 = Ψ * ∂f 1 (0n) △ Ψ * ∂f 2 (0n) = Ψ * ∂f 1 (0n) △ ∂f 2 (0n) ,
the last equality being a particular case of [41, Theorem 5.2].

Remark . A word of caution is in order. The vertical sum

K p 1 ⊕ v K p 2 = {(x, t) ∈ R n+1 : ∥x∥ p 1 + ∥x∥ p 2 ≤ t}
of two ℓ p -cones is not necessarily an ℓ p -cone, and the vertical sum

E(Q 1 ) ⊕ v E(Q 2 ) = {(x, t) ∈ R n+1 : √ ⟨x, Q 1 x⟩ + √ ⟨x, Q 2 x⟩ ≤ t}
of two elliptic cones is not necessarily an elliptic cone. The same remark applies to horizontal sums.

Summarizing, if K 1 and K 2 are epigraphical cones, then so are their Minkowski sum, their intersection, their vertical sum, and their horizontal sum. Furthermore,

f K 1 +K 2 = f K 1 f K 2 f K 1 ∩K 2 = f K 1 ∨ f K 2 f K 1 ⊕vK 2 = f K 1 + f K 2 f K 1 ⊕ h K 2 = f K 1 △f K 2 .
In the same vein one can derive composition rules for yet more elaborated operations like direct and inverse addition of order p ∈]1, ∞[ (in the sense of [41, Definition 3.1]).

The dual of an epigraphical cone

Duality plays a conspicuous role throughout this work. Recall that the dual cone of a closed convex cone K in R d is defined by

K + = {v ∈ R d : ⟨v, z⟩ ≥ 0 for all z ∈ K}.
The first question that comes to mind is this: is the dual of an epigraphical cone yet another epigraphical cone? Before answering this question, we recall the concept of polarity for positive sublinear functions. That a sublinear function is positive means that it is nonnegative and vanishes only at the origin. Lyubich [START_REF] Lyubich | Perron-Frobenius theory for finite-dimensional spaces with a hyperbolic cone[END_REF] refers to a positive sublinear function as a subnorm and to the epigraph of a subnorm as an hyperbolic cone. We do not adhere to this terminology because subnorms and hyperbolic cones are often used with a different meaning in the literature.

Recall that the polar f

• : R n → R of a positive sublinear function f : R n → R is given by f • (y) = max x̸ =0n ⟨y, x⟩ f (x) .
The skewed polar of f is the function

y ∈ R n → f 3 (y) = f • (-y).
The definition of polarity is classical and can be found in many books. Skewed polarity is less standard, but it appears already in [START_REF] Lyubich | Perron-Frobenius theory for finite-dimensional spaces with a hyperbolic cone[END_REF]. Of course, if f is even, then f • is also even and there is no distinction between polarity and skewed polarity. The next theorem is a particular case of [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF]Lemma 2.10], see also [START_REF] Rockafellar | Convex Analysis[END_REF]Theorem 14.4] or [START_REF] Lyubich | Perron-Frobenius theory for finite-dimensional spaces with a hyperbolic cone[END_REF]Proposition 3.1].

Theorem 2.6. Let f : R n → R be a positive sublinear function. Then f 3 : R n → R is a positive sublinear function and (epif ) + = epif 3 .

So, under the hypotheses of the above theorem, the dual of epif is again an epigraphical cone. In practice, computing the dual of an epigraphical cone amounts to computing the polar of a positive sublinear function.

Example 2.7. Elliptic cones and ℓ p -cones can both be embedded in the wider class of epigraphical cones of the form

Γ p,H := {(x, t) ∈ R n+1 : ∥Hx∥ p ≤ t} (5)
with p ∈ [1, ∞] and H denoting a nonsingular matrix of order n. The model ( 5) is quite flexible and encompasses a large variety of cones arising in applications. Note that f (x) = ∥Hx∥ p defines a positive sublinear function. Its polar is given by f • (y) = ∥Gy∥ q with p -1 +q -1 = 1 and G = (H -1 ) T . One gets in this way the formula (Γ p,H ) + = Γ q,G .

Degree of solidity of an epigraphical cone

The inradius of a nontrivial closed convex cone K in R d is defined as the number

ρ(K) := max x∈K∩S d dist[x, bd(K)] (6)
with dist[x, C] denoting the distance from x to a set C. The concept of inradius has been discussed in detail in [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] and also in earlier references like [START_REF] Belloni | A geometric analysis of Renegar's condition number, and its interplay with conic curvature[END_REF][START_REF] Belloni | On the second-order feasibility cone: primal-dual representation and efficient projection[END_REF][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] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF][START_REF] Iusem | Normality and modulability indices. Part I: Convex cones in normed spaces[END_REF]. Note that ρ(K) corresponds to the optimal value of the maximization problem maximize r (7)

∥x∥ = 1 r ∈ [0, 1] x + rB d ⊂ K.
Geometrically speaking, the optimization problem ( 7) is about finding a ball of largest radius centered at a unit vector and contained in K. This observation explains why the term (6) measures to which extent the cone K is solid. In fact, the function K → ρ(K) is a solidity index in the axiomatic sense of [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF].

The following lemma by Henrion and Seeger [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF] tells how to compute the inradius of an epigraphical cone. This result is not absolutely general because the associated nonnegative sublinear function is required to be even.

Lemma 3.1. If the nonnegative sublinear function

f : R n → R is even, then ρ(epif ) = [1 + β 2 f ] -1/2 with β f := max y∈∂f (0n) ∥y∥ = max ∥x∥=1 f (x).
With this lemma at hand one can derive a number of new results. For instance, one can obtain a rule for computing the inradius of an epigraphical cone whose associated nonnegative sublinear function has the decomposable structure

f (x) = N (φ(x 1 , . . . , x m ), ψ(x m+1 , . . . , x n )) (8)
with N standing for a norm in R 2 that is monotonic in the sense that

|a 1 | ≤ |c 1 | and |a 2 | ≤ |c 2 | imply N (a 1 , a 2 ) ≤ N (c 1 , c 2 ).
The next theorem involves the expression

N ⊗ (b 1 , b 2 ) := max a 2 1 +a 2 2 =1 N (a 1 b 1 , a 2 b 2 ),
which is yet another monotonic norm on R 2 , not to be confused with the dual norm of N .

Theorem 3.2. Let N be a monotonic norm in R 2 and let 2 ≤ m ≤ n -1. Suppose that the nonnegative sublinear functions φ : R m → R and ψ : R n-m → R are even. Then the inradius of the epigraphical cone

K N φ,ψ := {(x, t) ∈ R n+1 : N (φ(x 1 , . . . , x m ), ψ(x m+1 , . . . , x n )) ≤ t} is given by ρ(K N φ,ψ ) = [ 1 + ( N ⊗ (β φ , β ψ ) ) 2 ] -1/2 .
Proof. The function f given by ( 8) satisfies the hypotheses of Lemma 3.1. By splitting

x = (x 1 , . . . , x m u , x m+1 , . . . , x n v ) (9)
into two group of variables, one obtains

β f = max ∥u∥ 2 +∥v∥ 2 =1 N (φ(u), ψ(v)) = max a 2 1 +a 2 2 =1 a 1 ≥0,a 2 ≥0 max ∥u∥=a 1 ∥v∥=a 2 N (φ(u), ψ(v)) . ( 10
) But max ∥u∥=a 1 ∥v∥=a 2 N (φ(u), ψ(v)) = N ( max ∥u∥=a 1 φ(u), max ∥v∥=a 2 ψ(v) ) = N (a 1 β φ , a 2 β ψ ) ,
where the first equality is due to the monotonicity of N . That N is monotonic has further consequences: it implies that

N (c 1 , c 2 ) = N (|c 1 |, |c 2 |) for all (c 1 , c 2 ) ∈ R 2 .
Hence, the constraints a 1 ≥ 0, a 2 ≥ 0 in [START_REF] Chu | On a multivariate eigenvalue problem. I. Algebraic theory and a power method[END_REF] are superfluous. This shows that β f = N ⊗ (β φ , β ψ ) and completes the proof of the theorem.

The example below illustrates how Theorem 3.2 works in practice.

Example 3.3. Consider the convex cone

K = { (x, t) ∈ R n+1 : max 1≤k≤m |x k | + n ∑ k=m+1 |x k | ≤ t } .
Here φ is the ℓ ∞ -norm on R m , ψ is the ℓ 1 -norm on R n-m , and

N (c 1 , c 2 ) = |c 1 | + |c 2 |. Hence, N ⊗ (b 1 , b 2 ) = [b 2 1 + b 2 2 ] 1/2 , β φ = 1, β ψ = √ n -m.
One gets in this way

ρ(K) = [2 + n -m] -1/2 .
The monotonicity of N is an essential assumption in Theorem 3.2. For instance, a norm like

N (c 1 , c 2 ) = |c 1 | + |c 2 -c 1 |
would not be acceptable. On the other hand, Theorem 3.2 admits a more general formulation in which the vector x ∈ R n is split into several portions, and not just into two portions as in [START_REF] Burton | Contracting the maximal points of an ordered convex set[END_REF].

The next proposition provides a rule for computing the inradius of an intersection of epigraphical cones.

Proposition 3.4. If the nonnegative sublinear functions

f 1 , f 2 : R n → R are even, then ρ(epif 1 ∩ epif 2 ) = min{ρ(epif 1 ), ρ(epif 2 )}.
Proof. It is enough to combine (2), Lemma 3.1, and the fact that

β f 1 ∨f 2 = max{β f 1 , β f 2 }.
The next example illustrates the usefulness of Proposition 3.4. It concerns the intersection of two elliptic cones.

Example 3.5. The inradius of an elliptic cone E(Q) is related to the maximal eigenvalue of the corresponding matrix Q. Indeed, if one sets f (x) = √ ⟨x, Qx⟩, then

β f = max ∥x∥=1 √ ⟨x, Qx⟩ = √ λ max (Q) ,
and Lemma 3.1 yields

ρ(E(Q)) = [1 + λ max (Q)] -1/2 . ( 11
)
If Q 1 , Q 2 are two positive definite symmetric matrices of order n, then one gets

ρ(E(Q 1 ) ∩ E(Q 2 )) = [1 + max{λ max (Q 1 ), λ max (Q 2 )}] -1/2 .
Next we state a rule for computing the inradius of a Minkowski sum of epigraphical cones.

Proposition 3.6. If the nonnegative sublinear functions f 1 , f 2 : R n → R are even, then

ρ(epif 1 + epif 2 ) = [1 + β 2 ] -1/2
with β being the optimal value of the nonconvex optimization problem maximize {∥y∥ :

y ∈ ∂f 1 (0 n ) ∩ ∂f 2 (0 n )}. ( 12
)
If f 1 and f 2 are norms, then [START_REF] Fiedler | Cones which are topheavy with respect to a norm[END_REF] can be reformulated in the simpler form

maximize ∥y∥ f • 1 (y) ≤ 1 f • 2 (y) ≤ 1.
Proof. The combination of (3) and Lemma 3.1 yields

ρ(epif 1 + epif 2 ) = [1 + ∥C∥ 2 ] -1/2 ,
where ∥C∥ := max y∈C ∥y∥ and

C = ∂(f 1 f 2 )(0 n ) = ∂f 1 (0 n ) ∩ ∂f 2 (0 n ).
Suppose now that f 1 and f 2 are norms. In such a case, f • 1 and f • 2 are norms as well, and

∂f k (0 n ) = {f k ≤ 1} • = {f • k ≤ 1} for k ∈ {1, 2}.
Example 3.7. Let Q 1 , Q 2 be positive definite symmetric matrices of order n. Then

ρ(E(Q 1 ) + E(Q 2 )) = [1 + χ] -1/2
with χ standing for optimal value of the nonconvex quadratic optimization problem maximize ∥y∥ 2 (13)

⟨y, Q -1 1 y⟩ ≤ 1 ⟨y, Q -1 2 y⟩ ≤ 1.
This follows from Proposition 3.6 and the fact that

f • k (y) = [ ⟨y, Q -1 k y⟩ ] 1/2 .
The geometric interpretation of ( 13) is clear: one searches for a vector of largest norm that lies in the intersection of two ellipsoids. The optimization problem ( 13) is very interesting in itself and has been studied by a number of authors, see [START_REF] Arzelier | LMI approximations for the radius of the intersection of ellipsoids: survey[END_REF][START_REF] Henrion | Algebraic approach to robust controller design: a geometric interpretation[END_REF][START_REF] Henrion | Rank-one LMI approach to simultaneous stabilization of linear systems[END_REF][START_REF] Nemirovski | On maximization of quadratic form over intersection of ellipsoids with common center[END_REF][START_REF] Papavassilopoulos | The diameter of an intersection of ellipsoids and BMI robust synthesis[END_REF] and references therein.

Proposition 3.8. If the nonnegative sublinear functions f 1 , f 2 : R n → R are even, then

ρ(epif 1 ⊕ v epif 2 ) = [1 + β 2 v ] -1/2 ρ(epif 1 ⊕ h epif 2 ) = [1 + β 2 h ] -1/2 with β v = max{∥y∥ : y ∈ ∂f 1 (0 n ) + ∂f 2 (0 n )} β h = max{∥y∥ : y ∈ ∂f 1 (0 n ) △ ∂f 2 (0 n )}.
Proof. It is a matter of combining Proposition 2.5 and Lemma 3.1.

Remark . If f 1 , f 2 are norms, then β v is the optimal value of the nonconvex optimization problem

maximize ∥u + v∥ f • 1 (u) ≤ 1 f • 2 (v) ≤ 1.

Intermezzo: a tale of multispectra

As mentioned before, the inradius of an elliptic cone E(Q) is related to the maximal eigenvalue of the corresponding matrix Q. We now explain how the formula (11) can be extended to the context of a vertical sum

K Q := E(Q 1 ) ⊕ v . . . ⊕ v E(Q N ) = { (x, t) ∈ R n+1 : N ∑ k=1 √ ⟨x, Q k x⟩ ≤ t }
of finitely many elliptic cones. Here Q = {Q 1 , . . . , Q N } is a collection of positive definite symmetric matrices of order n.

The next proposition provides a formula for computing ρ(K Q ). Before stating such result in a proper manner, we need first to open a parenthesis and recall some facts on multispectra. The so-called Multivariate Eigenvalue Problem (MEP) consists in finding real numbers λ 1 , . . . , λ N such that the linear system

     A 1,1 A 1,2 . . . A 1,N A 2,1 A 2,2 . . . A 2,N . . . . . . . . . . . . A N,1 A N,2 . . . A N,N           ξ 1 ξ 2 . . . ξ N      =      λ 1 ξ 1 λ 2 ξ 2 . . . λ N ξ N      (14)
admits a solution (ξ 1 , . . . , ξ N ) ∈ (R n ) N satisfying the N -fold normalization condition

∥ξ 1 ∥ = 1, . . . , ∥ξ N ∥ = 1. ( 15
)
The multispectrum of the block structured matrix

A = [A i,j ] i,j∈{1,...,N } ( 16
)
is denoted by msp(A) and is defined as the set of N -tuples (λ 1 , . . . , λ N ) for which the system ( 14)-( 15) is solvable.

A concrete MEP for a symmetric block structured matrix ( 16) was introduced for the first time by Hotelling [START_REF] Hotelling | The most predictable criterion[END_REF]. The specific problem treated by Hotelling concerns the determination of canonical correlation coefficients for multivariate statistics. An iterative method for solving MEP's was proposed by Horst [START_REF] Horst | Relations among m sets of measures[END_REF]. For additional information on theoretical aspects and algorithms for solving MEP's, the reader is conveyed to the references [START_REF] Amri | Spectral analysis of coupled linear complementarity problems[END_REF][START_REF] Chu | On a multivariate eigenvalue problem. I. Algebraic theory and a power method[END_REF][START_REF] Hanafi | Global optimality of the successive Maxbet algorithm[END_REF][START_REF] Horst | Factor Analysis of Data Matrices[END_REF]. The only thing one needs to know here about multispectra is the next lemma. Its proof is a simple matter of applying the technique of Lagrange multipliers and proceeding as in [2, Proposition 2.2].

Lemma 3.9 (Variational Principle for Multispectra). If the block structured matrix (16) is symmetric, then msp(A) is nonempty and the maximal value of the quadratic form

q A (ξ) = ⟨    ξ 1 . . . ξ N    , A    ξ 1 . . . ξ N    ⟩ on the multisphere (S n ) N is equal to max { N ∑ k=1 λ k : (λ 1 , . . . , λ N ) ∈ msp(A) } .
We now are ready to state:

Proposition 3.10. Let Q = {Q 1 , . . . , Q N } be a finite collection of positive definite symmetric matrices of order n. Then ρ(K Q ) = [1 + χ Q ] -1/2 ( 17
)
with

χ Q = max { N ∑ k=1 λ k : (λ 1 , . . . , λ N ) ∈ msp(A Q ) } ( 18
)
and A Q denoting the symmetric block structured matrix whose (i, j)-block is given by Q

1/2 i Q 1/2 j .
Proof. Proposition 3.8 extends to the vertical sum of N elliptic cones and yields ( 17) with χ Q denoting the optimal value of

χ Q = max    N ∑ k=1 η k 2 : ⟨η k , Q -1 k η k ⟩ ≤ 1 for all k ∈ {1, . . . , N }    . ( 19
)
The inequality constraints in [START_REF] Hanafi | Global optimality of the successive Maxbet algorithm[END_REF] are all are active at a solution. This fact and the change of variables

ξ k = Q -1/2 k η k lead to the equivalent formulation χ Q = max    N ∑ k=1 Q 1/2 k ξ k 2 : ∥ξ 1 ∥ = 1, . . . , ∥ξ N ∥ = 1    . ( 20
)
Note that

N ∑ k=1 Q 1/2 k ξ k 2 = N ∑ i=1 N ∑ j=1 ⟨ξ i , Q 1/2 i Q 1/2 j ξ j ⟩ = ⟨    ξ 1 . . . ξ N    , A Q    ξ 1 . . . ξ N    ⟩
is the quadratic form associated to A Q . Finally, passing from ( 20) to ( 18) is a matter of applying Lemma 3.9.

Degree of pointedness of an epigraphical cone

As explained 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], there are many ways of measuring the degree of pointedness of a nontrivial closed convex cone K in R d . One can use for instance the coefficient

µ(K) := min z∈co(K∩ S d ) ∥z∥ ( 21
)
whose geometric meaning is clear: the minimization problem ( 21) is about finding the least norm element of the convex set co(K ∩ S d ). One refers to the number µ(K) as the basic pointedness coefficient of K. General information on the function K → µ(K) can be found scattered in a number of references [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][START_REF] Iusem | Normality and modulability indices. Part I: Convex cones in normed spaces[END_REF]. By definition, the basic pointedness coefficient of epif is given by

µ(epif ) = min { √ ∥x∥ 2 + t 2 : (x, t) ∈ co [epif ∩ S n+1 ] } . ( 22
)
Some simplification is achieved in [START_REF] Henrion | On properties of different notions of centers for convex cones[END_REF] if the nonnegative sublinear function f is even. Indeed, under such assumption one obtains the simpler expression

µ(epif ) = min {t : (0 n , t) ∈ co [epif ∩ S n+1 ]} . ( 23
)
Obtaining an explicit characterization of the convex hull of epif ∩ S n+1 is however a hard task. The next result tells how to compute the minimal value [START_REF] Henrion | Inradius and circumradius of various convex cones arising in applications[END_REF]

µ(epif ) = α f √ 1 + α 2 f ( 24
)
with

α f := min ∥x∥=1 f (x). ( 25
)
Proof. Consider first the case α f = 0, that is, f (x) = 0 for some x ∈ S n . Since f is even, one also has f (-x) = 0. Hence, epif is not pointed because it contains the line R (x, 0). In short, both sides of ( 24) are equal to zero. Consider now the case α f ̸ = 0, that is, f vanishes only at 0 n . By Proposition 2.1 one knows already that epif is pointed. Recall that f is also even. Thus, f and f • are norms. On the other hand, according to a duality result stated in [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF]Proposition 6.3], one can write µ(K) = ρ(K + ) for any nontrivial closed convex cone K in R d . Hence, by combining Theorem 2.6 and Lemma 3.1 one gets

µ(epif ) = ρ((epif ) + ) = [1 + γ 2 f ] -1/2
with γ f = max ∥y∥=1 f • (y). The rest of the proof consists in showing that γ f = 1/α f , but this equality is known or ought to be known. In fact, it follows from the general identity

min x̸ =0 g(x) h(x) = min y̸ =0 h • (y) g • (y) (26)
that applies to any pair g, h of norms on R n .

The next two corollaries are given only with a sketch of proof because everything is more or less the same as in Section 3.

Corollary 4.2. If the nonnegative sublinear functions

f 1 , f 2 : R n → R are even, then µ(epif 1 + epif 2 ) = min{µ(epif 1 ), µ(epif 2 )}. (27)
Proof. If α f 1 or α f 2 is equal to 0, then epif 1 or epif 2 is not pointed. In such a case the Minkowski sum epif 1 + epif 2 is not pointed either. Hence, both sides of ( 27) are equal to 0. Suppose now that α f 1 and α f 2 are both different from 0. In such a case, f 1 and f 2 are norms. One gets

µ(epif 1 + epif 2 ) = ρ ( (epif 1 + epif 2 ) + ) = ρ (epif • 1 ∩ epif • 2 ) = min {ρ (epif • 1 ) , ρ (epif • 2 )} = min {µ (epif 1 ) , µ (epif 2 )} .
We have omitted some easy intermediate steps.

Corollary 4.3. If the nonnegative sublinear functions

f 1 , f 2 : R n → R are even, then µ(epif 1 ∩ epif 2 ) = α √ 1 + α 2 with α = min x∈Sn max{f 1 (x), f 2 (x)}.
If f 1 and f 2 are norms, then one can also write

µ(epif 1 ∩ epif 2 ) = [1 + β 2 ] -1/2
with β standing for the optimal value of the nonconvex optimization problem maximize ∥x∥

f 1 (x) ≤ 1 f 2 (x) ≤ 1.
Proof. The first formula is obtained by applying Lemma 4.1 to f 1 ∨ f 2 . Unfortunately, the number α is not always easy to evaluate. If f 1 and f 2 are norms, then one can proceed a bit further by using duality arguments. More precisely, one can write

µ(epif 1 ∩ epif 2 ) = ρ ( (epif 1 ∩ epif 2 ) + ) = ρ (epif • 1 + epif • 2 )
and then one can apply Proposition 3.6 to the pair f

• 1 , f • 2 .
Example 4.4. Let Q 1 , Q 2 be positive definite symmetric matrices of order n. Then

µ(E(Q 1 ) ∩ E(Q 2 )) = [1 + χ] -1/2
with χ standing for optimal value of the nonconvex quadratic optimization problem

maximize ∥x∥ 2 ⟨x, Q 1 x⟩ ≤ 1 ⟨x, Q 2 x⟩ ≤ 1.
5 Angular structure of an epigraphical cone

Maximal angle

The maximal angle of a nontrivial closed convex cone K in R d is defined as the number

θ max (K) := max z,v∈K∩ S d arccos ⟨z, v⟩. (28)
One of the reasons why the maximal angle of a convex cone is a mathematical tool of interest is that the coefficients

γ(K) := 1 - θ max (K) π ν(K) := cos ( θ max (K) 2
) also serve to measure the degree of pointedness of K. Both of them qualify as pointedness index in the axiomatic sense of [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF]. The maximal angle function K → θ max (K) has however many other uses.

One says that (y, z) is an antipodal pair of K if y and z are unit vectors in K achieving the maximal angle of the cone, i.e., y, z ∈ K ∩ S d and arccos ⟨z, v⟩ = θ max (K). Antipodality in convex cones has been extensively theorized in [START_REF] Iusem | On pairs of vectors achieving the maximal angle of a convex cone[END_REF][START_REF] Iusem | Antipodal pairs, critical pairs, and Nash angular equilibria in convex cones[END_REF][START_REF] Iusem | Searching for critical angles in a convex cone[END_REF]. The value [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF] has been computed in [START_REF] Gourion | Critical angles in polyhedral convex cones: numerical and statistical considerations[END_REF][START_REF] Hiriart-Urruty | A variational approach to copositive matrices[END_REF][START_REF] Iusem | Angular analysis of two classes of non-polyhedral convex cones: the point of view of optimization theory[END_REF] for several particular convex cones arising in applications. The case of an epigraphical cone is discussed next.

Suppose for a moment that f : R n → R is a positive sublinear function, so that B f is compact. If one looks at the formula (1), then the first idea that comes to mind is that finding an antipodal pair of epif is somewhat related to the problem of finding two points in B f that are at maximal distance. Unfortunately, solving the diameter maximization problem

diam(B f ) = max u,v∈B f ∥u -v∥ (29)
is of no use, and, what is worse, it may lead to wrong conclusions. Since

(epif ) ∩ S n+1 = { (x, 1) √ 1 + ∥x∥ 2 : x ∈ B f } , what we have to compute is cos [θ max (epif )] = min u,v∈B f ⟨u, v⟩ + 1 √ 1 + ∥u∥ 2 √ 1 + ∥v∥ 2 .
This can be reformulated as

√ 2 √ 1 -cos [θ max (epif )] = max u,v∈B f d (u, v) (30)
with

d (u, v) := (u, 1) √ 1 + ∥u∥ 2 - (v, 1) √ 1 + ∥v∥ 2 .
One has to evaluate the "diameter" of B f after all, but with respect to the metric d and not with respect to the usual metric of R n . The next example shows that a solution to [START_REF] Iusem | On pairs of vectors achieving the maximal angle of a convex cone[END_REF] is not necessarily a solution to [START_REF] Iusem | Measuring the degree of pointedness of a closed convex cone: a metric approach[END_REF]. Example 5.1 concerns a function f that is not even. The next theorem tells how to compute the maximal angle of epif when f is even. Theorem 5.2. Suppose that the nonnegative sublinear function f : R n → R is even. Then

θ max (epif ) = 2 arccos   α f √ 1 + α 2 f   = arccos ( α 2 f -1 α 2 f + 1 ) . ( 31
)
The maximal angle of epif is formed with the unit vectors

1 √ 1 + α 2 f (x, α f ) and 1 √ 1 + α 2 f (-x, α f ) , ( 32
)
where x ∈ S n is any solution to the minimization problem [START_REF] Horst | Relations among m sets of measures[END_REF].

Proof. The second equality in [START_REF] Iusem | Angular analysis of two classes of non-polyhedral convex cones: the point of view of optimization theory[END_REF] follows from a general trigonometric identity, namely, the formula for the cosine of the half-angle. If α f = 0, then the three terms in [START_REF] Iusem | Angular analysis of two classes of non-polyhedral convex cones: the point of view of optimization theory[END_REF] are equal to π and this angle is attained with the vectors (x, 0) and (-x, 0). Suppose then that α f ̸ = 0, in which case f is a norm. Let ϑ f denote the maximal angle that a unit vector of epif forms with respect to the canonical vector e n+1 . Hence, cos ϑ f is equal to the optimal value of the minimization problem minimize ⟨0 n , u⟩ + 1t (u, t) ∈ epif

∥u∥ 2 + t 2 = 1.
Since this minimum is attained on the boundary of epif , the constraint (u, t) ∈ epif can be converted into f (u) = t. By getting rid of the variable t, one obtains cos ϑ f = min {f (u) :

∥u∥ 2 + [f (u)] 2 = 1}.
A positive homogeneity argument leads to

cos ϑ f = min u̸ =0n f (u) √ ∥u∥ 2 + [f (u)] 2 = min u̸ =0n [ 1 + ( ∥u∥ f (u) ) 2 ] -1/2 = [ 1 + ( min u̸ =0n f (u) ∥u∥ ) -2 ] -1/2 .
We have shown in this way that

cos ϑ f = ( 1 + α -2 f ) -1/2 = α f / ( 1 + α 2 f ) 1/2 .
The evenness of f has not been used yet. This property is needed for proving the following claim:

θ max (epif ) = 2ϑ f . ( 33
)
From the very definition of ϑ f one sees that epif is contained in the revolution cone

rev(ϑ f ) = { (u, t) ∈ R n+1 : (cos ϑ f ) √ ∥u∥ 2 + t 2 ≤ t } = {(x, t) ∈ R n+1 : (tan ϑ) -1 ∥x∥ ≤ t}
whose axis is generated by e n+1 and whose half-aperture angle is equal to ϑ f . Hence,

θ max (epif ) ≤ θ max (rev(ϑ f )) = 2ϑ f .
On the other hand, both vectors in [START_REF] Iusem | Normality and modulability indices. Part I: Convex cones in normed spaces[END_REF] have unit length and belong to epif . Therefore

θ max (epif ) ≥ arccos ( ⟨x, -x⟩ + α 2 f 1 + α 2 f ) = arccos ( α 2 f -1 α 2 f + 1 ) = 2θ f .
This confirms the claim [START_REF] Iusem | Antipodal pairs, critical pairs, and Nash angular equilibria in convex cones[END_REF] and completes the proof.

Remark . If f is not even, then the second vector in [START_REF] Iusem | Normality and modulability indices. Part I: Convex cones in normed spaces[END_REF] does not belong necessarily to epif . It is still possible to write

θ max (epif ) ≤ 2 arccos   α f √ 1 + α 2 f   ,
but this inequality may be very coarse. To see this one must construct an epigraphical cone whose associated sublinear function is not even, but highly skewed.

By combining Lemma 3.1 and Theorem 5.2 one gets the following by-product.

Corollary 5.3. If the nonnegative sublinear function f : R n → R is even, then ν(epif ) = µ(epif ).

Minimal angle

The maximal angle of a convex cone is one side of the story. The other side is the minimal angle.

In fact, between the maximal and the minimal one there is a full collection of intermediate critical angles.

Definition 5.4. Let K be a closed convex cone of R d and let z, v be unit vectors in K.

i) That (z, v) is a critical pair of K means that v -⟨z, v⟩z ∈ K + and z -⟨z, v⟩v ∈ K + .
ii) The angle θ(z, v) = arccos ⟨z, v⟩ formed by a critical pair is called a critical angle. A critical pair (u, v) and the corresponding critical angle θ(u, v) are said to be proper if u and v are not collinear.

iii) The angular spectrum of K, indicated with the symbol asp(K), is the set of all proper critical angles of K. The smallest element of this set is denoted by θ min (K) and called the minimal angle of K.

The criticality conditions formulated in (i) can be seen as first order optimality conditions for the angle maximization problem [START_REF] Iusem | Axiomatization of the index of pointedness for closed convex cones[END_REF]. General information concerning the theory of critical angles in convex cones can be found in the standard references [START_REF] Iusem | On pairs of vectors achieving the maximal angle of a convex cone[END_REF][START_REF] Iusem | Antipodal pairs, critical pairs, and Nash angular equilibria in convex cones[END_REF][START_REF] Iusem | Searching for critical angles in a convex cone[END_REF].

The next proposition provides a formula for computing the minimal angle of an epigraphical cone associated to a norm. Proposition 5.5. Let f be a norm on R n . Then

θ min (epif ) = 2 arcsin   1 √ 1 + β 2 f   = arcsin ( 2β f 1 + β 2 f ) . ( 34
)
Proof. In the present context the cone epif is solid and pointed. In view of a duality result for extremal angles established in [START_REF] Iusem | Searching for critical angles in a convex cone[END_REF]Theorem 3], one can write

θ min (epif ) = π -θ max ((epif ) + ). (35)
By exploiting [START_REF] Lee | A volume problem for an n-dimensional ellipsoid intersecting with a hyperplane[END_REF] and Theorems 2.6 and 5.2, one obtains

θ min (epif ) = π -θ max (epif • ) = π -2 arccos   δ f √ 1 + δ 2 f   (36)
Example 5.9. Let Q be a positive definite symmetric matrix of order n. As pointed out in [34, Lemma 1], a nonzero vector u ∈ R n satisfies the relation

⟨u, Q -1 u⟩⟨u, Qu⟩ = ∥u∥ 4
if and only if u is an eigenvector of Q. Hence, the umbilical points of the norm f = Ψ * E(Q) are the eigenvectors of Q, and

usp(Ψ * E(Q) ) = { √ ⟨u, Qu⟩ ∥u∥ : u is an eigenvector of Q } = { √ λ : λ is an eigenvalue of Q } .
This norm admits at most n umbilical values.

The next proposition shows that the umbilical points of a positive sublinear function f are exactly the eigenvectors of the subdifferential map ∂f . A nonzero vector u ∈ R n is called an eigenvector of a multivalued map A : R n ⇒ R n if there exists a scalar λ ∈ R such that λu ∈ A(u).

In such a case one refers to λ as an eigenvalue of A associated to the eigenvector u. Proposition 5.10. Let f : R n → R be a positive sublinear function. Then u is an umbilical point of f if and only if u is an eigenvector of ∂f .

Proof. We suppose that u ̸ = 0 n , otherwise there is nothing to prove. That u is an umbilical point of f amounts to saying that u solves the maximization problem

f • (u) = max x̸ =0n ⟨u, x⟩ f (x)
.

Equivalently, u solves the convex optimization problem

f • (u) = max f (x)≤f (u) ⟨u, x⟩ f (u) . ( 37
)
The so-called Slater qualification hypothesis holds for [START_REF] Moreau | Fonctionelles Convexes, Equations aux derivées partielles[END_REF]. Hence, for u to solve [START_REF] Moreau | Fonctionelles Convexes, Equations aux derivées partielles[END_REF] it is necessary and sufficient that u f (u) ∈ η ∂f (u) [START_REF] Nemirovski | On maximization of quadratic form over intersection of ellipsoids with common center[END_REF] for some Karush-Kuhn-Tucker multiplier η ≥ 0. Since the left-hand side of (38) is a nonzero vector, the multiplier η must be positive. Thus, [START_REF] Nemirovski | On maximization of quadratic form over intersection of ellipsoids with common center[END_REF] can be written in the form λu ∈ ∂f (u) with λ = [ηf (u)] -1 . This proves the proposition.

We now are ready to state: (c) x is an umbilical point of f .

Proof. We start with the "if" part. Suppose that the system (a)-(c) is in force. We must prove that the vectors (x, t) = (x, f (x)) (39) (y, s) = (-x, f (x)) [START_REF] Rockafellar | Convex Analysis[END_REF] form a proper critical pair of epif . These vectors have unit length because

[f (x)] 2 + ∥x∥ 2 = 1. (41)
Clearly, (39) belongs to epif . The vector (40) also belongs to epif because f is even. Since [START_REF] Papavassilopoulos | The diameter of an intersection of ellipsoids and BMI robust synthesis[END_REF] and ( 40

) 43
Thanks to Theorem 2.6 and the fact that f is even, the system ( 42)-( 43) reduces to

(1 + λ)f • (x) ≤ (1 -λ)f (x). ( 44
)
Since x is an umbilical point of f , there exists a positive scalar γ such that f (x) = γ∥x∥ and f • (x) = (1/γ)∥x∥.

In fact, γ is the umbilical value of f associated to x. Hence, the inequality (44) becomes

(1 + λ) ≤ γ 2 (1 -λ). ( 46
)
By combining [START_REF] Seeger | Direct and inverse addition in convex analysis and applications[END_REF] and ( 45) one gets ∥x∥ 2 = (1 + γ 2 ) -1 . Hence,

1 + λ = 2(1 -∥x∥ 2 ) = 2γ 2 /(1 + γ 2 ) γ 2 (1 -λ) = 2γ 2 ∥x∥ 2 = 2γ 2 /(1 + γ 2 ).
This proves that (46) holds in fact as an equality. We now prove the "only if" part. We assume that (x, t) and (y, s) form a critical pair of epif . By criticality, we have

(

Example 5 . 1 .

 51 Let f : R 2 → R be the positive sublinear function whose set B f is the triangle of vertices a = (100, 0), b = (0, -100), and c = (-40, 1). The diameter of B f is 100 √ 2 and this value is achieved with the pair (a, b). However, this pair does not solve the maximization problem (30) because d (a, c)

Theorem 5 . 11 .

 511 Let f be a norm on R n . The vectors (x, t) and (y, s) form a proper critical pair of epif if and only if the following three conditions hold:(a) y = -x. (b) s = t = √ 1 -∥x∥ 2 = f (x).

  ) are not collinear, their inner productλ = ⟨(x, f (x)), (-x, f (x))⟩ = [f (x)] 2 -∥x∥ 2 = 1 -2∥x∥ 2 belongs to the open interval ] -1, 1[. It remains to show that (x, f (x)) -λ (-x, f (x)) ∈ (epif ) +(42)(-x, f (x)) -λ (x, f (x)) ∈ (epif ) + . (

Corollary 2.2. Let

  The following intuitive result is part of the folklore on convex cones. It can be obtained as a consequence of Proposition 2.1 and [1, Theorem 1. 48]. f : R n → R be a nonnegative sublinear function vanishing only at 0 n . Then (x, 1) is a generator of epif if and only if x is an extreme point of B f .

  in practice. One can see Lemma 4.1 as a sort of dual version of Lemma 3.1.

Lemma 4.1. If the nonnegative sublinear function f : R n → R is even, then

  Since f is even, both (-x, t) and (-y, s) also belong to epif . Multiplying the left hand side of (47) by (-x, t), one gets + t 2 -λ(ts -⟨x, y⟩).

	0 ≤ = -∥x∥ 2 By plugging the value of λ and rearranging, one obtains ⟨ (-x, t), (x, t) -λ(y, s) ⟩
	0 ≤ t

x, t) -λ(y, s) ∈ (epif ) + (47) (y, s) -λ(x, t) ∈ (epif ) +

(48)

with λ = ⟨x, y⟩ + ts. 2 (1 -s 2 ) -∥x∥ 2 + ⟨x, y⟩ 2 . (49)

with δ f = min ∥y∥=1 f • (y). But the identity [START_REF] Horst | Factor Analysis of Data Matrices[END_REF] shows that δ f = 1/β f . By plugging this value in [START_REF] Lyubich | Perron-Frobenius theory for finite-dimensional spaces with a hyperbolic cone[END_REF] and simplifying one arrives at the first equality in [START_REF] Iusem | Searching for critical angles in a convex cone[END_REF]. The second equality in [START_REF] Iusem | Searching for critical angles in a convex cone[END_REF] is a general trigonometric identity.

We state without proof two immediate corollaries. ) .

Corollary 5.7. Let N be a monotonic norm in R 2 and let 2 ≤ m ≤ n -1. Let φ and ψ be norms on R m and R n-m , respectively. Then the minimal angle of the epigraphical cone K N φ,ψ is given by

and sin

In particular,

Critical angles

Identifying the critical angles of a convex cone is in general a difficult task. The next theorem tell how to construct critical pairs in an epigraphical cone associated to a norm. The key ingredient of the discussion is the concept of umbilical point for a positive sublinear function f : R n → R. From the very definition of polar function it follows that

Of course, any positive multiple of an umbilical point is an umbilical point. The umbilical spectrum of f is the set

Each element of this set is called an umbilical value of f .

Recall that (x, t) and (y, s) are vectors of unit length, i.e.,

The combination of ( 49) and (50) produces

and therefore

By the same token, multiplying the left hand side of ( 48) by (-y, s) leads to

By combining (51) and (52) one gets

Hence, y = ±x and t = s. The case y = x must be ruled out because we are assuming properness of the critical pair {(x, t), (y, s)}. We conclude that y = -x, establishing (a). The first equality in (b) has also been proved. The second one is contained in (50), and the third one follows from the fact that (x, t) is necessarily in the boundary of epif . Next we prove (c). By using (47) and the parts (a) and (b), one gets the inequality (44) and

These three relations together yield 2 . This proves that x is an umbilical point of f . The proof of Theorem 5.11 is very much in the spirit of [START_REF] Iusem | Searching for critical angles in a convex cone[END_REF]Theorem 5], but the novelty of our approach is the introduction and use of the concept of umbilicity. As a complement to Theorem 5.11 we provide below a full description of the angular spectrum of epif . Theorem 5.12. If f is a norm on R n , then

Proof. Let γ be an umbilical value of f . Then γ = f (u)/∥u∥ for some umbilical point u of f . The "if" part of Theorem 5.11 shows that the vectors

form a proper critical pair of epif . Hence, the corresponding angle

is a proper critical angle of epif . This proves the "⊃" part of the equality (53). Conversely, let θ be a proper critical angle of epif . Suppose that θ is formed with the proper critical pair {(x, t), (y, s)}.

Due to the "only if" part of Theorem 5.11, one must have (54)-(55) for some umbilical point u of f . Hence, θ has the form (56) with γ = f (u)/∥u∥. This completes the proof of (53).

Remark . If f is a norm on R n , then the angular spectrum of epif can also be represented as

Since (f • ) • = f , the norms f and f • have the same umbilical points and asp(epif

Note that θ is a proper critical angle of epif • if and only if π -θ is a proper critical angle of epif .

It is worthwhile to mention that (53) can be inverted so as to get a characterization for the umbilical spectrum of a norm:

In practice, the relation (57) is less interesting than (53) because asp(epif ) is usually hard to compute. However, (57) has some theoretical relevance. For instance, since a polyhedral cone is known to have finitely many critical angles, one gets:

Corollary 5.13. A polyhedral norm on R n has finitely many umbilical values.

By way of conclusion

This completes the first part of our work on epigraphical cones. The companion paper [START_REF] Seeger | Epigraphical cones II[END_REF] presents additional material on this topic: smoothness and rotundity in epigraphical cones, facial analysis, etc. Applications of epigraphical cones in optimization theory are also considered. In particular, [START_REF] Seeger | Epigraphical cones II[END_REF] provides rules for constructing barrier functions for epigraphical cones.