François Fillastre
email: francois.fillastre@u-cergy.fr

Stephanie Alexander

Thierry Barbot

Francesco Bonsante

Bruno Colbois

Ivan Izmestiev

Yves Martinez-Maure

Joan Porti

Jean-Marc Schlenker

Graham Smith

Rolf Schneider

Fuchsian convex bodies: basics of Brunn-Minkowski theory

The hyperbolic space H d can be defined as a pseudo-sphere in the pd 1q Minkowski space-time. In this paper, a Fuchsian group Γ is a group of linear isometries of the Minkowski space such that H d {Γ is a compact manifold. We introduce Fuchsian convex bodies, which are closed convex sets in Minkowski space, globally invariant for the action of a Fuchsian group. A volume can be associated to each Fuchsian convex body, and, if the group is fixed, Minkowski addition behaves well. Then Fuchsian convex bodies can be studied in the same manner as convex bodies of Euclidean space in the classical Brunn-Minkowski theory. For example, support functions can be defined, as functions on a compact hyperbolic manifold instead of the sphere.

The main result is the convexity of the associated volume (it is log concave in the classical setting). This implies analogs of Alexandrov-Fenchel and Brunn-Minkowski inequalities. Here the inequalities are reversed.

ones, are simpler than in the Euclidean case.

Introduction

There are two main motivations behind the definitions and results presented here. See next section for a precise definition of Fuchsian convex bodies, the main object of this paper, and Fuchsian convex surfaces (boundaries of Fuchsian convex bodies).

The first motivation is to show that the geometry of Fuchsian convex surfaces in the Minkowski space is the right analogue of the classical geometry of convex compact hypersurfaces in the Euclidean space. In the present paper, we show the analogue of the basics results of what is called Brunn-Minkowski theory. Roughly speaking, the matter is to study the relations between the sum and the volume of the bodies under consideration. Actually here we associate to each convex set the volume of another region of the space, determined by the convex set, so we will call it the covolume of the convex set. This generalization is as natural as, for example, going from the round sphere to compact hyperbolic surfaces. To strengthen this idea, existing results can be put into perspective. Indeed, Fuchsian convex surfaces are not new objects. As far I know, smooth Fuchsian hypersurfaces appeared in [START_REF] Oliker | Codazzi tensors and equations of Monge-Ampère type on compact manifolds of constant sectional curvature[END_REF], see Subsection 3.3. The simplest examples of convex Fuchsian surfaces are convex hulls of the orbit of one point for the action of the Fuchsian group. They were considered in [START_REF] Näätänen | The convex hull construction for compact surfaces and the Dirichlet polygon[END_REF], in relation with the seminal papers [Penner, 1987, Epstein andPenner, 1988]. See also [START_REF] Charney | Nonpositively curved, piecewise Euclidean structures on hyperbolic manifolds[END_REF]. The idea is to study hyperbolic problems via the extrinsic structure given by the Minkowski space. For a recent illustration see [START_REF] Espinar | Hypersurfaces in H n 1 and conformally invariant equations: the generalized Christoffel and Nirenberg problems[END_REF]. The first study of Fuchsian surfaces for their own is probably [START_REF] Labourie | Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante[END_REF]. The authors proved that for any Riemannian metric on a compact surface of genus ¥ 2 with negative curvature, there exists an isometric convex Fuchsian surface in the 2 1-Minkowski space, up to a quotient. In the Euclidean case, the analog problem is known as Weyl problem. A uniqueness result is also given. This kind of result about realization of abstract metrics by (hyper)surfaces invariant under a group action seems to go back to former papers of F. Labourie and to [Gromov, 1986]. The polyhedral analog of [START_REF] Labourie | Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante[END_REF] is considered in [Fillastre, 2011a]. An important intermediate result, about polyhedral infinitesimal rigidity in d 2, was proved in [Schlenker, 2007] (Fuchsian analogue of Dehn theorem). More recently, a Fuchsian analogue of the "Alexandrov prescribed curvature problem" was proved in [Bertrand, 2010]. The proof uses optimal mass transport. A refinement of this result in the polyhedral d 2 case was obtained in [Iskhakov, 2000]. A solution for the Christoffel problem (prescribed sum of the radii of curvature in the regular case) for Fuchsian convex bodies will be given in [START_REF] Fillastre | The general Christoffel problem in Minkowski spacetime[END_REF] as well as for more general convex sets in the Minkowski space (with or without group action), similarly to [Lopes de Lima and Soares de Lira, 2006].

The second motivation is that, up to a quotient, the results presented here are about the covolume defined by convex Cauchy surfaces in the simplest case of flat Lorentzian manifolds, namely the quotient of the interior of the future cone by a Fuchsian group. It is relevant to consider them in a larger class of flat Lorentzian manifolds, known as maximal globally hyperbolic Cauchy-compact flat spacetimes. They were considered in the seminal paper [Mess, 2007], see [START_REF] Andersson | Notes on: "Lorentz spacetimes of constant curvature[END_REF] and [Barbot, 2005[START_REF] Bonsante | Flat spacetimes with compact hyperbolic Cauchy surfaces[END_REF]. Roughly speaking, one could consider hypersurfaces in the Minkowski space invariant under a group of isometries whose set of linear isometries forms a Fuchsian group (translations are added). In d 2, for such smooth strictly convex surfaces, a Minkowski theorem (generalizing Theorem 3.8 in this dimension) was proved recently in [START_REF] Béguin | Prescribing Gauss curvature of surfaces in 3-dimensional spacetimes application to the Minkowski problem in the Minkowski space[END_REF]. Maybe some of the basic objects introduced in the present paper could be extended to the point to these manifolds.

The paper is organized as follows. Section 2 introduces, among main definitions, the tool to study (Fuchsian) convex bodies, the support functions. The case of the C 2 Fuchsian convex bodies (roughly speaking, the ones with a sufficiently regular boundary) is treated in Section 3 and the one of polyhedral Fuchsian convex bodies in Section 4. These two sections are independent. In Section 5 the general results are obtained by polyhedral approximation. It appears that the proofs of the main results, even though very analogous to the classical 2 Definitions

Fuchsian convex bodies

The Minkowski space-time of dimension pd 1q, d ¥ 1, is R d 1 endowed with the symmetric bilinear form xx, yy ¡ x 1 y 1 ¤ ¤ ¤ x d y d ¡ x d 1 y d 1 .

We will denote by F the interior of the future cone of the origin. It is the set of future time-like vectors: the set of x such that xx, xy ¡ 0 (time-like) and the last coordinate of x for the standard basis is positive (future). The pseudo-sphere contained in F at distance t from the origin of R d 1 is

H d t tx R d 1 |xx, xy ¡ ¡t 2 , x d 1 ¡ 0u.
All along the paper we identify H d 1 with the hyperbolic space H d . In particular the isometries of H d are identified with the linear isometries of the Minkowski space keeping H d 1 invariant [Benedetti and Petronio, 1992, A.2.4].

Note that for any point x F , there exists t such that x H d t .

Definition 2.1. A Fuchsian group is a subgroup of the linear isometries group of R d 1 , fixing setwise F and acting freely cocompactly on H d (i.e. H d {Γ is a compact manifold).

A Fuchsian convex body is the data of a convex closed proper subset K of F , together with a Fuchsian group Γ, such that ΓK K. A Γ-convex body is a Fuchsian convex body with Fuchsian group Γ.

A Fuchsian convex surface is the boundary of a Fuchsian convex body.

A Fuchsian convex body has to be thought as the analogue of a convex body (compact convex set), with the compactness condition replaced by a "cocompactness" condition (we will see that a Fuchsian convex body is never bounded). Joan Porti pointed out to the author that what is done in this paper is probably true without the requirement that the group has no torsion.

We will adapt the classical theory to the Fuchsian case. For that one we mainly follow [Schneider, 1993].

Examples

The simplest examples of Fuchsian convex surfaces are the H d t (note that all Fuchsian groups act freely and cocompactly on H d t ). Their convex sides are Fuchsian convex bodies, denoted by B d t , and B d 1 is sometimes denoted by B d or B. This example shows that a given convex set can be a Fuchsian convex body for many Fuchsian groups.

Given a Fuchsian group Γ, we will see in the remaining of the paper two ways of constructing convex Fuchsian bodies. First, given a finite number of points in F , the convex hull of their orbits for Γ is a Fuchsian convex body, see Subsection 4.1, where a dual construction is introduced. Second, we will see in Subsection 3.1 that any function on the compact hyperbolic manifold H d {Γ satisfying a differential relation corresponds to a Fuchsian convex body. Hence the question of examples reduces to the question of finding the group Γ, that implies to find compact hyperbolic manifolds. Standard concrete examples of compact hyperbolic manifolds can be easily found in the literature about hyperbolic manifolds. For a general construction in any dimension see [START_REF] Gromov | Nonarithmetic groups in Lobachevsky spaces[END_REF].

Notwithstanding it is not obvious to get explicit generators. Of course the case d 1 is totally trivial as a Fuchsian group is generated by a boost ¢ cosh t sinh t sinh t cosh t , for a non-zero real t. For d 2, explicit generators can be constructed following [Maskit, 2001]. For Figure 2 and a computation at the end of the paper (the figure comes from a part of a Fuchsian convex body that can be manipulate on the author's webpage), the group is the simpliest acting on H 2 , namely the one having a regular octagon as fundamental domain in a disc model. Generators are given in [Katok, 1992].

Remark on the signature of the bilinear form The classical theory of convex bodies uses the usual scalar product on R d 1 . Here we used the usual bilinear form of signature pd, 1q. A natural question is to ask what happens if we consider a bilinear form of signature pd 1 ¡ k, kq. (Obviously, the vector structure, the volume, the Levi-Civita connection (and hence the geodesics), the topology and the notion of convexity don't depend on the signature. Moreover, any linear map preserving the bilinear form is of determinant one, hence preserves the volume.)

Let us consider first the case of the usual bilinear form with signature pd ¡ 1, 2q (d ¥ 3). The set of vectors of pseudo-norm ¡1 is a model of the Anti-de Sitter space, which is the Lorentzian analogue of the Hyperbolic space. First of all, we need groups of linear isometries acting cocompactly on the Anti-de Sitter space. They exist only in odd dimensions [START_REF] Barbot | Group actions on Lorentz spaces, mathematical aspects: a survey[END_REF]. Moreover, Anti-de Sitter space does not bound a convex set.

Finally, another interest of the present construction is that, as noted in the introduction, some objects introduced here could serve to study some kind of flat Lorentzian manifolds (with compact Cauchy surface), which can themselves be related to some problems coming from General Relativity. It is not clear if as many attention is given to pseudo-Riemannian manifolds with different signatures.

Support planes

For a subset A of R d 1 , a support plane of A at x is an hyperplane H with x A H and A entirely contained in one side of H. Lemma 2.2. Let K be a Γ-convex body. Then (i) K is not contained in a codimension ¡ 0 plane.

(ii) K is future convex:

(a) through each boundary point there is a support plane;

(b) all support planes are space-like; (c) K is contained in the future side of its support planes.

Proof. By definition K is not empty. Let x K. As K F , there exists a t such that x H d t , and by definition, all the elements of the orbit Γx of x belong to K H d t . Suppose that K is contained in a codimension ¡ 0 hyperplane H. Then there would exist a codimension 1 hyperplane H I with H H I , and Γx H I H d t . This means that on H d t (which is homothetic to the hyperbolic space for the induced metric), Γx is contained in a totally geodesic hyperplane, a hypersphere or a horosphere (depending on H I to be time-like, space-like or light-like), that is clearly impossible. Schneider, 1993, 1.3.2].

(i) is proved. (ii)(a) is a general property of convex closed subset of R d 1 [
Let x K and let H be the support plane of K at x. where d is the hyperbolic distance, are bounded [Ratcliffe, 2006]. The sublemma is a characteristic property of discrete sets with bounded Dirichlet regions [Charney et al., 1997, Lemma 3].

Lemma 2.4. Let K be a Γ-convex body and x K. For any λ ¥ 1, λx K.

Proof. From the definition of K, it is not hard to see that it has non empty interior. And as K is closed, if the lemma was false, there would exist a point on the boundary of K and a support plane at this point such that x in its past, that is impossible because of Lemma 2.2.

Let us recall the following elementary results, see e.g. [Ratcliffe, 2006, 3.1.1,3.1.2 is the support hyperplane of A at x with inward normal η.

Lemma 2.6. Let K be a Γ-convex body. For any future time-like vector η, suptxx, ηy ¡ |x Ku exists, is attained at a point of K and is negative. In particular any future time-like vector η is an inward normal of K.

A future time-like vector η is the inward normal of a single support hyperplane of K.

Proof. From (i) of Lemma 2.5, txx, ηy ¡ |x Ku is bounded from above by zero hence the sup exists. The sup is a negative number, as a sufficiently small translation of the vector hyperplane H orthogonal to η in direction of F does not meet K. This follows from the separation theorem [Schneider, 1993, 1.3.4], because the origin is the only common point between H and the boundary of F . As K is closed, the sup is attained when the parallel displacement of H meets K. Suppose that two different support hyperplanes of K have the same inward normal. Hence one is contained in the past of the other, that is impossible.

Support functions

Let K be a Γ-convex body. The extended support function H of K is dη F , Hpηq suptxx, ηy ¡ |x Ku.

(2)

We know from Lemma 2.6 that it is a negative function on F . As an example the extended support function of

B d t is equal to ¡t ¡xη, ηy ¡ . Definition 2.7. A function f : A Ñ R on a convex subset A of R d 1 is sublinear (on A) if it is positively homogeneous of degree one: dη A, f pληq λ f pηq dλ ¡ 0, (3)
and subadditive:

dη, µ A, f pη µq ¤ f pηq f pµq.

(4)

A sublinear function is convex, in particular it is continuous (by assumptions it takes only finite values in A). (It is usefull to note that for a positively homogeneous of degree one function, convexity and sublinearity are equivalent.) It is straightforward from the definition that an extended support function is sublinear and Γ-invariant. It is useful to expand the definition of extended support function to the whole space. The total support function of a Γ-convex body K is dη R d 1 , Hpηq suptxx, ηy ¡ |x Ku.

(5)

We will consider the total support function for any convex subset of R d 1 . The infinite value is allowed. We have the following important property, see [Hörmander, 2007, Theorem 2.2.8].

Proposition 2.8. Let f be a lower semi-continuous, convex and positively homogeneous of degree one function on R d 1 (the infinite value is allowed). The set

F tx R d 1 |xx, ηy ¡ ¤ f pηq dη R d 1 u is a closed convex set with total support function f .
From the definition we get: Lemma 2.9. A convex subset of R d 1 is a point if and only if its total support function is a linear form. (If the point is p, the linear form is x¤, py ¡ .)

In particular, the total support function of a Fuchsian convex body is never a linear form.

The relation between the extended support function and the total support function is as follows.

Lemma 2.10. The total support function H of a Γ-convex body with extended support function H is equal to:

• H on F , • 0 on the future light-like vectors and at 0,

• V elsewhere.
Moreover H is a Γ invariant sublinear function.

Proof. We have the following cases

• If η is future time-like then Hpηq Hpηq.

• If η is past time-like or past light-like, then by (i) of Sublemma 2.5 for x K, xx, ηy ¡ ¡ 0, and by Lemma 2.4, Hpηq V.

• If η is space-like, as seen in the proof of (ii)(b) of Lemma 2.2, there exists points of K on both side of the orthogonal (for x¤, ¤y ¡ ) of η. Hence there exists x K with xx, ηy ¡ ¡ 0, and by the preceding argument, Hpηq V. If H is invariant under the action of Γ, then H is negative or H 0 on F , and H 0 on fF .

Note that H 0 is the support function of (the closure of) F . Proof. Let be a future light-like vector. As Γ acts cocompactly on H d , there exists a sequence of γ k Γ such that for any future time-like ray r, the sequence γ n r converges to the ray containing [Ratcliffe, 2006, Example 2, 12.2]. From this sequence we take a sequence γ k η for a future time-like vector η. We have Hpγ k ηq Hpηq.

From this sequence we take a sequence of vectors η I k which all have the same pd 1qth coordinate pη I k q d 1 as , say d 1 (hence η I k Ñ ). We have η I k d 1 {pγ k ηq d 1 γ k η, and by homogeneity Hpη I k q Hp d 1 ηq{pγ k ηq d 1 that goes to 0 as k goes to infinity (pγ k ηq d 1 goes to infinity). This proves t H 0 on fF as for any fF ¦ and any η F , HK p q lim tÓ0 H K p tpx ¡ qq (see for example Theorem 7.5 in [Rockafellar, 1997]). In the same way we get that Hp0q 0.

As H is convex and equal to 0 on fF , it is non-positive on F . Suppose that there exists x F with Hpxq 0, and let y F ztxu. By homogeneity, Hpλxq 0 for all λ ¡ 0. Up to choose an appropriate λ, we can suppose that the line joining x and y meets fF in two points. Let be the one such that there exists λ s0, 1r such that x λ p1 ¡ λqy. By convexity and because Hpxq Hp q 0, we get 0 ¤ Hpyq, hence Hpyq 0.

Lemma 2.12. Let H be a negative sublinear Γ-invariant function on F . The set

K tx F |xx, ηy ¡ ¤ Hpηq dη F u is a Γ-convex body with extended support function H.
Proof. Let H be as in Sublemma 2.11. From Proposition 2.8, the set

K tx R d 1 |xx, ηy ¡ ¤ Hpηq dη R d 1 u is a closed convex set, with total support function H. Let us see that K K.
As Hpηq V outside the closure F of the future cone we have

K tx R d 1 |xx, ηy ¡ ¤ Hpηq dη F u. For η F , Hpηq ¤ 0, it follows that K is contained in F : K tx F |xx, ηy ¡ ¤ Hpηq dη F u.
As H is Γ-invariant, H and K are Γ-invariant too. For x K fF , the origin is an accumulating point of Γx from Sublemma 2.13. So for any η F , Hpηq, which is the sup of xx, ηy ¡ for x K, should be zero, that is false. Hence K tx F |xx, ηy ¡ ¤ Hpηq dη F u and as Hpηq 0 on fF we get

K tx F |xx, ηy ¡ ¤ Hpηq dη F u K.
The remainder is easy.

Sublemma 2.13. Let Γ be a Fuchsian group and let x be a future light-like vector. Then the origin is an accumulating point of Γx.

Proof. Suppose it is false. As Γ acts cocompactly on H d , there exists an horizontal space-like hyperplane S such that a fundamental domain on H d for the action of Γ lies below S . If the origin is not an accumulating point, then there exists λ ¡ 0 such that the horoball ty

H d | ¡ 1 ¤ xλx, yy ¡ 0u
and its images for the action of Γ remain above S . This contradicts the definition of fundamental domain.

The

polar dual K ¦ of a Γ-convex body K is, if H is the extended support function of K: K ¦ tx F |Hpxq ¤ ¡1u. For example, pB d t q ¦ B d 1{t .
It is not hard to see that K ¦ is a Γ-convex body, and that K ¦¦ K (see the convex bodies case [Schneider, 1993, 1.6.1]). Moreover the points of the boundary of K ¦ are the ¡1

Hpηq η for η H d . The inverse of this map is the projection f pxq x c ¡xx,xy ¡ . Hence, exchanging the roles of K and K ¦ , we get that the projection of a Fuchsian convex body along rays from the origin gives is a homeomorphism between fK and H d .

Minkowski sum and covolume

The (Minkowski) addition of two sets A, B R d 1 is defined as

A B : ta b|a A, b Bu.
It is well-known that the addition of two convex sets is a convex set. Moreover the sum of two future time-like vectors is a future time-like vector, in particular it is never zero. So the sum of two Γ-convex bodies is contained in F and closed [Rockafellar and Wets, 1998, 3.12]. As a Fuchsian group Γ acts by linear isometries, the sum is a Γ-convex body, and the space KpΓq of Γ-convex bodies is invariant under the addition. Note also that KpΓq is invariant under multiplication by positive scalars. It is straightforward to check that extended support functions behave well under these operations:

H K L H K H L , K, L KpΓq, H λK λH K , λ ¡ 0, K KpΓq.
Note also that from the definition of the extended support function,

K L ô H K ¤ H L .
Identifying Γ-convex bodies with their support functions, KpΓq is a cone in the vector space of homogeneous of degree 1, continuous, real, Γ-invariant, functions on F . By homogeneity this corresponds to a cone in the vector space of continuous real Γ-invariant functions on H d F , and to a cone in the vector space of continuous real functions on the compact hyperbolic manifold H d {Γ. A function in one of this two last cones is called a support function.

Let K KpΓq. Its covolume covolpKq is the volume of pF zKq{Γ (for the Lebesgue measure of R d 1 ). It is a finite positive number and covolpλKq λ d 1 covolpKq.

Note that

K L ñ covolpKq ¥ covolpLq.
As defined above, the covolume of a Γ-convex body K is the volume of a compact set of R d 1 , namely the volume of the intersection of F zK with a fundamental cone for the action of Γ. For such compact (non-convex) sets there is a Brunn-Minkowski theory, see for example [Gardner, 2002]. See also [START_REF] Bahn | A Brunn-Minkowski type theorem on the Minkowski spacetime[END_REF]. But this does not give results about covolume of Γ-convex bodies. The reason is that, for two Γ-convex bodies K 1 and K 2 , F zpK 1 K 2 q (from which we define the covolume of K 1 K 2 ) is not equal to pF zK 1 q pF zK 2 q. For example in d 1,

¡ 0 1{2 © ¡ 5{8 9{8
© pF zB F zBq but does not belong to F zpB Bq.

C 2 case

The first subsection is an adaptation of the classical case [Schneider, 1993]. The remainder is the analog of [Alexandrov, 1938] (in [Alexandrov, 1996]). See also [START_REF] Bonnesen | Theory of convex bodies[END_REF], [Leichtweiß, 1993], [Hörmander, 2007], [Busemann, 2008], and [START_REF] Guan | A form of Alexandrov-Fenchel inequality[END_REF] for a kind of extension.

The objects and results in this section which can be defined intrinsically on a hyperbolic manifold are already known in more generality, see [START_REF] Oliker | Codazzi tensors and equations of Monge-Ampère type on compact manifolds of constant sectional curvature[END_REF] and the references therein. See also Subsection 3.3.

Regularity of the support function

Differentiability Let K be a Γ-convex body with extended support function H, and let η F . From Lemma 2.6 there exists a unique support hyperplane H of K with inward normal η.

Lemma 3.1. The intersection F of H and K is reduced to a single point p if and only if H is differentiable at η F . In this case p ∇ η H (the gradient for x¤, ¤y ¡ of H at η).

Proof. As H is convex all one-sided directional derivatives exist [Schneider, 1993, p. 25]. Let us denote such derivative in the direction of u R d 1 at the point η by d η Hpuq. The proof of the lemma is based on the following fact:

The function R d 1 u Þ Ñ d η Hpuq is the total support function of F.
Indeed, if H is differentiable at η, the fact says that the total support function of F is a linear form, and from Lemma 2.9, F is a point. Conversely, if F is a point p, from Lemma 2.9 its total support function is a linear form, hence partial derivatives of H exist and as H is convex, this implies differentiability [Schneider, 1993, 1.5.6]. Moreover for all u R d 1 , xp, uy ¡ d η Hpuq. [Schneider, 1993, 1.5.4], Proposition 2.8 applies and d η H is the total support function of

Now we prove the fact. The function

d η H is sublinear on R d 1
F I tx R d 1 |xx, uy ¡ ¤ d η Hpuq du R d 1 u.
We have to prove that F I F. Let H be the extension of H to R d 1 . By definition of directional derivative, the sublinearity of H gives d η H ¤ H. From the proof of Lemma 2.12, this implies that F I K. In particular, for y F I , xy, ηy ¡ ¤ Hpηq. On the other hand y F I implies xy, ¡ηy ¡ ¤ d η Hp¡ηq ¡Hpηq (the last equality follows from the definition of directional derivative, using the homogeneity of H). Then xy, ηy ¡ Hpηq so y H, hence F I F H K.

Let y F. By definition xy, ηy ¡ Hpηq and for any w F , xy, wy ¡ ¤ Hpwq. For sufficiently small positive λ and any u R d 1 , w η λu is future time-like and xy, uy ¡ ¤ Hpη λuq ¡ Hpηq λ so when λ Ñ 0 we have xy, uy ¡ ¤ d η Hpuq hence F F I . The fact is proved.

If the extended support function H of a Γ-convex body K is differentiable, the above lemma allows to define the map Gpηq ∇ η H from F to fK R d 1 . This can be expressed in term of h, the restriction of H to H d . We use "hyperbolic coordinates" on F : an orthonormal frame on H d extended to an orthonormal frame of F with the decomposition r 2 g H d ¡ d r 2 of the metric on F . ∇ η H has d 1 entries, and, at η H d , the d first ones are the coordinates of ∇ η h (here ∇ is the gradient on H d ). We identify ∇ η h T η H d R d 1 with a vector of R d 1 . The last component of ∇ η H is ¡fH{frpηq, and, using the homogeneity of H, it is equal to ¡hpηq when η H d . Note that at such a point,

T η F is the direct sum of T η H d and η. It follows that, for η H d , ∇ η H ∇ η h ¡ hpηqη. (6)
This has a clear geometric interpretation, see Figure 1. We denote by G the restriction of G to H d and by W its differential (the reversed shape operator). If T ν is the hyperplane of R d 1 orthogonal to ν H d for x¤, ¤, y ¡ , W is considered as a map from T ν to T ν . We get from (6), or from the equation above, the Gauss formula and the 1-homogeneity of H, using again hyperbolic coordinates on F :

W i j p∇ 2 hq i j ¡ hδ i j , (7)
with ∇ 2 the second covariant derivative (the Hessian) on H d , δ i j the Kronecker symbol and h the restriction of H to H d . In particular W is symmetric, and its real eigenvalues r 1 , . . . , r d are the radii of curvature of K. Taking the trace on both parts of the equation above leads to

r 1 ¤ ¤ ¤ r d ∆ H d h ¡ dh (8)
where ∆ H d is the Laplacian on the hyperbolic space. It is easy to check that, for γ Γ, ∇ γη H γ∇ η H and

D 2 γη H D 2 η H.
In particular the objects introduced above can be defined on H d {Γ.

C 2 body Let K be a Γ-convex body. The Gauss map N is a multivalued map which associates to each x in the boundary of K the set of unit inward normals of K at x, which are considered as elements of

H d . If the boundary of K is a C 2 hypersurface and if the Gauss map is a C 1 -homeomorphism from the boundary of K to H d , K is C 2 .
In this case we can define the shape operator B ∇N, which is a self-adjoint operator. Its eigenvalues are the principal curvatures κ i of K, and they are never zero as B has maximal rank by assumption. As K is convex, it is well-known that its principal curvatures are non-negative, hence they are positive. (This implies that K is actually strictly convex.)

Lemma 3.2. Under the identification of a Γ-convex body with its support function, the set of C 2 Γ-convex body is C 2 pΓq, the set of negative C 2 functions h on M H d {Γ such that pp∇ 2 hq i j ¡ hδ i j q ¡ 0 (9) (positive definite) for any orthonormal frame on M.

It follows that in the C 2 case G N ¡1 , W B ¡1 , and r i 1 7) and ( 9). Moreover G is surjective by Lemma 2.6. It follows that fK is C 1 . This implies that each point of fK has a unique support plane [Schneider, 1993, p. 104], i.e that the map G is injective. Finally it is a C 1 homeomorphism.

κ i ¥ N ¡1 . Proof. Let K be a C 2 Γ-convex
Let K ¦ be the polar dual of K. We know that the points on the boundary of K ¦ are graphs above H d as they have the form η{p¡hpηqq for η H d . Hence fK ¦ is C 2 as h is. Moreover the Gauss map image of the point η{p¡hpηqq of fK ¦ is Gpηq{ ¡xGpηq,Gpηqy ¡ : the Gauss map of K ¦ is a C 1 homeomorphism. It follows that K ¦ is C 2 . In particular its support function is C 2 . Repeating the argument, it follows that the boundary of

K ¦¦ K is C 2 .
To simplify the matter in the following, we will restrict ourselves to smooth (C V ) support functions, al- though this restriction will be relevant only in Subsection 3.4. We denote by C V pΓq the subset of smooth elements of C 2 pΓq. It corresponds to C V Γ-convex bodies, i.e. Γ-convex bodies with smooth boundary and with the Gauss map a C 1 diffeomorphism (hence smooth).

Lemma 3.3. C V pΓq is a convex cone and C V pΓq ¡ C V pΓq C V pΓq (any smooth function on H d {Γ is the difference of two functions of C V pΓq). Proof. It is clear that C V pΓq is a convex cone. Let h 1 C V pΓq and Z C V pΓq. As H d {Γ is compact, for t sufficiently large, Z th 1 satisfies (9) and is a negative function, hence there exists h 2 C V pΓq such that Z th 1 h 2 .

Covolume and Gaussian curvature operator

Let K be a C 2 Γ-convex body and let PpKq be F minus the interior of K. As PpKq{Γ is compact, the divergence theorem gives The Gaussian curvature (or Gauss-Kronecker curvature) κ of K is the product of the principal curvatures.

We will consider the map κ ¡1 which associates to each h C V pΓq the inverse of the Gaussian curvature of the convex body supported by h:

κ ¡1 phq d ¹ i1
r i phq (7) det p∇ 2 hq i j ¡ hδ i j ¨.

(11)

As the curvature is the Jacobian of the Gauss map, we get

pd 1qcovolpKq ¡ » M hκ ¡1 phq d M
where d M is the volume form on M H d {Γ. Finally let us consider the covolume as a functional on C V pΓq, which extension to the whole C V pΓq is immediate:

covolpXq ¡ 1 d 1 pX, κ ¡1 pXqq, X C V pΓq (12)
with p¤, ¤q the scalar product on L 2 pMq.

We will consider C V pΓq as a Fréchet space with the usual seminorms

}f} n n i1 sup xM |∇ i f pxq|,
with ∇ i the i-th covariant derivative and |¤| the norm, both given by the Riemannian metric of M. All derivatives will be directional (or Gâteaux) derivatives in Fréchet spaces as in [Hamilton, 1982]:

D Y covolpXq lim tÑ0 covolpY tXq ¡ covolpYq t , X, Y C V pΓq. (13)
Lemma 3.4. The function covol is C V on C V pΓq, and for h C V pΓq, X, Y C V pΓq, we have:

D h covolpXq ¡pX, κ ¡1 phqq, (14)
D 2
h covolpX, Yq ¡pX, D h κ ¡1 pYqq.

(15)

Moreover ( 14) is equivalent to pX, D h κ ¡1 pYqq pY, D h κ ¡1 pXqq.

(16)

Proof. The second order differential operator κ ¡1 is smooth as the determinant is smooth [Hamilton, 1982, 3.6.6]. Differentiating (12) we get

D h covolpXq ¡ 1 d 1 pX, κ ¡1 phqq ph, D h κ ¡1 pXqq ¨, (17)
but the bilinear form p¤, ¤q is continuous for the seminorms } ¤ } n (recall that it suffices to check continuity in each variable [Rudin, 1991, 2.17]). It follows that covol is C 1 , and by iteration that it is C V .

If ( 14) is true we get (15), and this expression is symmetric as covol is C 2 , so (16) holds.

Let us suppose that ( 16) is true. From (11), κ ¡1 is homogeneous of degree d, that gives D h κ ¡1 phq dκ ¡1 phq. Using this in (16) with Y h gives dpX, κ ¡1 phqq ph, D h κ ¡1 pXqq.

Inserting this equation in (17) leads to (14).

A proof of ( 16) is done in [START_REF] Cheng | On the regularity of the solution of the n-dimensional Minkowski problem[END_REF] (for the case of C 2 functions on the sphere). See also [START_REF] Oliker | Codazzi tensors and equations of Monge-Ampère type on compact manifolds of constant sectional curvature[END_REF] and reference therein for more generality. We will prove ( 14) following [Hörmander, 2007].

From the definition of κ ¡1 , the map D h κ ¡1 p¤q is linear, hence from (17) D h covolp¤q is also linear, so by Lemma 3.3 it suffices to prove (14) for X h I C V pΓq. We denote by K (resp. K I ) the Γ-convex body supported by h (resp. h I ) and by N (resp. N I ) its Gauss map. We have, for η F , ε ¡ 0, hpηq εh I pηq xη, N ¡1 pηq εpN I q ¡1 pηqy ¡ i.e h εh I supports the hypersurface with position vector N ¡1 pηq εpN I q ¡1 pηq.

For

a compact U R d , if f : U Ñ R d 1 is a local parametrization of fK, let us introduce F : U ¢ r0, εs Ñ R d 1 , py, tq Þ Ñ f pyq tpN I q ¡1 pNpfpyqqq.
It is a local parametrization of the set between the boundary of K and the boundary of K εK I . Locally, its covolume (which corresponds to covolph εh I q ¡ covolphq) is computed as

» FpU¢r0,εsq d vol » ε 0 » U |JacF| d y d t. ( 18
)
The Jacobian of F is equal to

¡ pN I q ¡1 pNpfpyqqq, f f fy 1 , . . . , f f fy d © tR
where R is a remaining term, and its determinant is equal to the determinant of

¡ pN I q ¡1 pNpfpyqqq, f f fy 1 , . . . , f f fy d ©
plus t times remaining terms. As p f f fy 1 , . . . , f f fy d q form a basis of the tangent hyperplane of fK, and as N is normal to this hyperplane, the determinant is equal to xpN I q ¡1 pNpfpyqqq, Np f pyqqy ¡ h I pNpfpyqqq times |Jacf|, plus t times remaining terms. The limit of (18) divided by ε when ε Ñ 0 gives

» U h I pNpfpyqq|Jacf| d y » f pUq h I pNq d fK.
The result follows by decomposing the boundary of K with suitable coordinate patches.

The main result of this section is the following.

Theorem 3.5. The second derivative of covol : C V pΓq Ñ R is positive definite. In particular the covolume of C V Γ-convex bodies is strictly convex.

Let us have a look at the case d 1. In this case κ ¡1 r, the unique radius of curvature. We parametrize the branch of the hyperbola by psinh t, cosh tq, and h becomes a function from R to R ¡ . Then (8) reads κ ¡1 phqptq ¡hptq h P ptq, and, as h is Γ-invariant, we can consider κ ¡1 as a linear operator on the set of C V functions on r0, s, if is the length of the circle H 1 {Γ. Using integration by parts and the fact that h is -periodic, we get

D 2 h covolph, hq ¡ph, κ ¡1 phqq ¡ » 0 hκ ¡1 phq » 0 ph 2 h I2 q.
We will prove a more general version of Theorem 3.5 in the next section, using the theory of mixed-volume. The proof is based on the following particular case.

Lemma 3.6. Let h 0 be the support function of B d (i.e. h 0 pηq ¡1). Then D 2 h 0 covol is positive definite.

Proof. Let X C V pΓq. From the definition (11) of κ ¡1

D h κ ¡1 pXq κ ¡1 phq d i1 r ¡1
i phqD h r i pXq 13 and as r i ph 0 q 1, D h 0 κ ¡1 pXq d i1 D h 0 r i pXq.

Differentiating (8) on both side at h 0 and passing to the quotient, the equation above gives

D h 0 κ ¡1 pXq ¡dX ∆ M X,
where ∆ M is the Laplacian on M H d {Γ. From (15), D 2 h 0 covolpX, Xq dpX, Xq ¡ p∆ M X, Xq, which is positive by property of the Laplacian, as M H d {Γ is compact.

Smooth Minkowski Theorem

One can ask if, given a positive function f on a hyperbolic compact manifold M H d {Γ, it is the Gauss curvature of a C 2 convex Fuchsian surface and if the former one is unique. By Lemma 3.2 and definition of the Gauss curvature, the question reduces to know if there exists a (unique) function h on M such that, in an orthogonal frame on M, f detpp∇ 2 hq i j ¡ hδ i j q and pp∇ 2 hq i j ¡ hδ i j q ¡ 0.

This PDE problem is solved in [START_REF] Oliker | Codazzi tensors and equations of Monge-Ampère type on compact manifolds of constant sectional curvature[END_REF] in the smooth case. Their main result (Theorem 3.4) can be written as follows.

Theorem 3.7. Let Γ be a Fuchsian group, f :

H d Ñ R be a positive C V Γ-invariant function.
There exists a unique C V Γ-convex body with Gauss curvature f .

Mixed curvature and mixed-covolume

The determinant is a homogeneous polynomial of degree d, and we denote by detp¤, . . . , ¤q its polar form, that is the unique symmetric d-linear form such that detpA, . . . , Aq detpAq for any d ¢ d symmetric matrix A (see for example Appendix A in [Hörmander, 2007]). We will need the following key result.

Theorem 3.8 ( [Alexandrov, 1996, p. 125]). Let A, A 3 . . . , A d be positive definite d ¢ d matrices and Z be a symmetric matrix. Then detpZ, A, A 3 , . . . , A d q 0 ñ detpZ, Z, A 3 , . . . , A d q ¤ 0, and equality holds if and only if Z is identically zero.

For any orthonormal frame on M H d {Γ and for X k C V pΓq, let us denote X P k : p∇ 2 X k q i j ¡ X k δ i j and let us introduce the mixed curvature κ ¡1 pX 1 , . . . , X d q : detpX P 1 , . . . , X P d q.

As covolpXq ¡ 1 d 1 pX, κ ¡1 pXqq, covol is a homogeneous polynomial of degree d 1. Its polar form covolp¤, . . . , ¤q (pd 1q entries) is the mixed-covolume.

Proof. It is formally self-adjoint because of the symmetry of the mixed-covolume. It is clearly second order linear. Let Z C V pΓq. From properties of the mixed determinant [Alexandrov, 1996, p. 121], κ ¡1 pZ, h 1 , . . . , h d¡1 q can be written, for an orthonormal frame on M, d i , j1

detph P 1 , . . . , h P d¡1 q i j p∇ 2 Zq i j ¡ Zδ i j ẅhere detph P 1 , . . . , h P d¡1 q i j is, up to a constant factor, the mixed determinant of the matrices obtained from the h P k by deleting the ith row and the jth column. Let us consider local coordinates on M around a point p such that at p, κ ¡1 pZ, h 1 , . . . , h d¡1 q has the expression above. By definition of C V pΓq, h P k are positive definite at p and then at p d i , j1

detph P 1 , . . . , h P d¡1 q i j x i x j is positive definite [Alexandrov, 1996, Lemma II p. 124].

Lemma 3.13. For any h 1 , . . . , h d¡1 in C V pΓq, the symmetric bilinear form covolp¤, ¤, h 1 , . . . , h d¡1 q has trivial kernel.

Proof. Suppose that Z belongs to the kernel of covolp¤, ¤, h 1 , . . . , h d¡1 q. As p¤, ¤q is an inner product, Z belongs to the kernel of κ ¡1 p¤, h 1 , . . . , h d¡1 q:

detpZ P , h P 1 , . . . , h P d¡1 q 0.

As h P i are positive definite matrices, by definition of C V pΓq, Theorem 3.8 implies that detpZ P , Z P , h P 2 , . . . , h P d¡1 q ¤ 0 so 0 covolpZ, Z, h 1 , . . . , h d¡1 q ¡ » M h 1 κ ¡1 pZ, Z, h 2 , . . . , h d¡1 q ¤ 0 but h 1 0 hence detpZ P , Z P , h P 2 , . . . , h P d¡1 q 0, and Theorem 3.8 says that Z P 0. Consider the 1-homogeneous extension Z of the Γ invariant map on H d defined by Z. From Subsection 3.1 it follows that the Hessian of Z in F is zero, hence that Z is affine. By invariance Z must be constant, and by homogeneity Z 0 hence Z 0.

Remark on Fuchsian Hedgehogs If we apply Cauchy-Schwarz inequality to the inner product of Theorem 3.11, we get a "reversed Alexandrov-Fenchel inequality" (see Theorem 5.5) for C V convex bodies, but also for any smooth function h on the hyperbolic manifold H d {Γ. From Lemma 3.3 there exist two elements h 1 , h 2 of C V pΓq with h h 1 ¡ h 2 . Hence h can be seen as the "support function" of the (maybe non convex) hypersurface made of the points ∇ η pH 1 ¡ H 2 q, η F . For example if h 1 and h 2 are the support functions of respectively B t 1 and B t 2 , then h is the support function of a pseudo-sphere in F if t 1 ¡ t 2 ¡ 0, of a point (the origin) if t 1 ¡ t 2 0 and of a pseudo-sphere in the past cone if t 1 ¡ t 2 0.

More generally, we could introduce "Fuchsian hedgehogs", whose "support functions" are difference of support functions of two Γ-convex bodies. They form the vector space in which the support functions of Γconvex bodies naturally live. In the Euclidean space, they were introduced in [START_REF] Langevin | Hérissons et multihérissons (enveloppes parametrées par leur application de Gauss)[END_REF]]. An Euclidean analog of the reversed Alexandrov-Fenchel inequality for smooth Fuchsian hedgehogs described above is done in [Martinez-Maure, 1999], among other results. It would be interesting to know if other results about hedgehogs have a Fuchsian analogue.

Polyhedral case

The classical analogue of this section comes from [Alexandrov, 1937] (see [Alexandrov, 1996]). See also [Schneider, 1993] and [Alexandrov, 2005]. The toy example d 1 is considered in the note [START_REF] Fillastre ; Fillastre | Polygons of the Lorentzian plane and spherical polyhedra[END_REF].

Support vectors

Definition of Fuchsian convex polyhedron The notation a u will represent the affine hyperplane over the vector hyperplane orthogonal to the vector a and passing through a: a u tx R d 1 |xx, ay ¡ xa, ay ¡ u.

(19) Definition 4.1. Let R pη 1 , . . . , η n q, n ¥ 1, with η i (pairwise non-collinear) vectors in the future cone F , and let Γ be a Fuchsian group. A Γ-convex polyhedron is the boundary of the intersection of the half-spaces bounded by the hyperplanes pγη i q u , dγ Γ, di 1, . . . , n, such that the vectors η i are inward pointing.

See Figure 2 for a simple example. Lemma 4.2. A Γ-convex polyhedron P (i) is a Γ-convex body, (ii) has a countable number of facets, (iii) is locally finite, (iv) each face is a convex Euclidean polytope.

Here convex polytope means convex compact polyhedron.

Proof. We denote by P i the Γ-convex polyhedron made from the vector η i and the group Γ. We will prove the lemma for P i . The general case follows because P is the intersection of a finite number of P i project along rays from the origin onto the vertices of the Dirichlet tessellation. In particular the vertices are in F , so P i F , because it is the convex hull of its vertices [Schneider, 1993, 1.4.3] and F is convex. In particular P i is a Γ-convex body due to Definition 2.1. And codimension k faces of P i projects onto codimension k faces of the Dirichlet tessellation, so P i is locally finite with a countable number of facets. Facets of P i are closed, as they project onto compact sets. In particular they are bounded as contained in F hence compact. They are convex polytopes by construction, and Euclidean as contained in space-like planes. Higher codimension faces are convex Euclidean polytopes as intersections of convex Euclidean polytopes.

Support numbers

The extended support function of a Γ-convex polyhedron P is piecewise linear (it is linear on each solid angle determined by the normals of the support planes at a vertex), it is why the data of the extended support function on each inward unit normal of the facets suffices to determine it. If η i is such a vector and h is the support function of P, we call the positive number hpiq : ¡hpη i q the ith support number of P.

The facet with normal η i is denoted by F i . Two adjacent facets F i and F j meet at a codimension 2 face F i j . If three facets F i , F j , F k meet at a codimension 3 face, then this face is denoted by F i jk . We denote by ϕ i j the hyperbolic distance between η i and η j , given by (see for example [Ratcliffe, 2006, (3.2.2)])

¡ cosh ϕ i j xη i , η j y ¡ . ( 20
)
Let p i be the foot of the perpendicular from the origin to the hyperplane H i containing the facet F i . In H i , let p i j be the foot of the perpendicular from p i to F i j . We denote by h i j the signed distance from p i to p i j : it is non negative if p i is in the same side of F j than P. See Figure 3.

For each i, h i j are the support numbers of the convex Euclidean polytope F i . (H i is identified with the Euclidean space R d , with p i as the origin.) If we denote by ω i jk the angle between p i p i j and p i p ik , it is wellknown that [Schneider, 1993, (5.1.3)]

h ik j h i j ¡ h ik cos ω i jk sin ω i jk . ( 21
)
We have a similar formula in Minkowski space [Fillastre, 2011b, Lemma 2.2]:

h i j ¡ hp jq ¡ hpiq cosh ϕ i j sinh ϕ i j . ( 22
)
In particular, If hpiq hp jq and if the quadrilateral is deformed under this condition, then fh i j

fh i j fhpjq ¡ 1 sinh ϕ i j , ( 23
)
fh i j fhpiq cosh ϕ i j sinh ϕ i j . ( 24
)
fhpiq cosh ϕ i j ¡ 1 sinh ϕ i j . ( 25
)
Space of polyhedra with parallel facets Let P be a Γ-convex polyhedron. We label the facets of P in a fundamental domain for the action of Γ. This set of label is denoted by I, and ΓI labels all the facets of P. Let R pη 1 , . . . , η n q be the inward unit normals of the facets of P labeled by I.

We denote by PpΓ, Rq the set of Γ-convex polyhedra with inward unit normals belonging to the set R. By identifying a Γ-convex polyhedron with its support numbers labeled by I, PpΓ, Rq is a subset of R n . (The corresponding vector of R n is the support vector of the polyhedron.) Note that this identification does not commute with the sum. Because the sum of two piecewise linear functions is a piecewise linear function, the Minkowski sum of two Γ-convex polyhedra is a Γ-convex polyhedron. (More precisely, the linear functions under consideration are of the form x¤, vy ¡ , with v a vertex of a polyhedron, hence a future time-like vector, and the sum of two future time-like vectors is a future time-like vector.) But even if the two polyhedra have parallel facets, new facets can appear in the sum. Later we will introduce a class of polyhedra such that the support vector of the Minkowski sum is the sum of the support vectors. Proof. The condition that the hyperplane supported by η j contains a facet of the polyhedron with support vector h can be written as hx R d 1 , di ΓI, i j, xη i , xy ¡ ¡hpiq and xη j , xy ¡ ¡hpjq.

By (20) PpΓ, Rq always contains the vector p1, . . . , 1q. The set is clearly open as a facet can't disappear for any sufficiently small deformation. It is also clearly invariant under homotheties of positive scale factor. So to prove that PpΓ, Rq is a convex cone it suffices to check that if h and h I belongs to PpΓ, Rq then h h I belongs to PpΓ, Rq. It is immediate from the above characterization.

Covolume of convex Fuchsian polyhedra

Let F be a facet of a Γ-convex polyhedron P, contained in a space-like hyperplane H, with support number h. For the induced metric, H is isometric to the Euclidean space R d , in which F is a convex polytope, with volume ApFq. We call ApFq the area of the facet. Let C be the cone in R d 1 over P with apex the origin. Its volume VpCq is invariant under the action of an orientation and time-orientation preserving linear isometry (they have determinant 1), hence to compute VpCq we can suppose that H is an horizontal hyperplane (constant last coordinate). For horizontal hyperplanes, the induced metric is the same if the ambient space is endowed with the standard Lorentzian metric or with the standard Euclidean metric. So the well-known formula applies:

VpCq 1 d 1 hApFq,
and then covolpPq 1 d 1 iI hpiqApF i q.

Identifying P with its support vector h,

if d ¤, ¤ h is the usual inner product of R n , we have covolphq 1 d 1 d h, Aphq h ( 26
)
where Aphq is the vector formed by the area of the facets ApF i q.

Lemma 4.4. The function covol is C 2 on R n , and for h PpΓ, Rq, X, Y R n , we have:

D h covolpXq d X, Aphq h , (27)
D 2 h covolpX, Yq d X, D h ApYq h . ( 28
)
Moreover (27) is equivalent to d X, D h ApYq h d Y, D h ApXq h . (29)
Proof. Let P be the polyhedron with support function h PpΓ, Rq. Let F i be a facet of P, with support numbers [Alexandrov, 2005, 8.2.3]

h i1 , . . . , h im . If V E is the d Euclidean volume, it is well-known that
fV E pF i q fh ik L ik (30)
where L ik is the area of the facet of F i with support number h ik (for d 1, one has1 instead of L ik ). ApF i q is not exactly as V E pF i q, because it is a function of h, and, when varying a hp jq, a new facet of F i can appear, as well as a new support number h i j of F i . Actually many new facets can appear, as many as hyperplanes with normals Γη j meeting F i . One has to consider F i as also supported by h i j (and eventually some orbits). In this case, L i j 0, and the variation of the volume is still given by formula (30). So even if the combinatorics of P changes under small change of a support number, there is no contribution to the change of the volume of the facets. So (30) gives

fApF i q fh ik L ik . (31)
We denote by

E j i ΓI is the set of indices k Γ j such that F k is adjacent to F i along a codimension 2
face. It can be empty. But for example if I is reduced to a single element i, E i i is the set of facets adjacent to F i along a codimension 2 face. If j Iztiu we get fApF i q fhpjq ķE j i fApF i q fh ik fh ik fhpjq

From ( 23) and (31) it follows that

fApF i q fhpjq ¡ ķE j i L ik sinh ϕ ik . (32)
For the diagonal terms:

fApF i q fhpiq jIztiu ķE j i fApF i q fh ik fh ik fhpiq ķE i i fApF i q fh ik fh ik fhpiq p31,24,25q jIztiu ķE j i cosh ϕ ik L ik sinh ϕ ik ķE i i L ik cosh ϕ ik ¡ 1 sinh ϕ ik . (33)
These expressions are continuous with respect to h, even if the combinatorics changes. So A is C 1 and from (26) covol is C 2 . If ( 27) is true, we get (28), and this expression is symmetric as covol is C 2 , so (29) holds. Let us suppose that ( 29 Let us prove (29). If e 1 , . . . , e n is the standard basis of R n , it suffices to prove (29) for X e i and Y e j , i j i.e that the gradient of A is symmetric. The sum in (32) means that, in fP{Γ, each times the ith polytope meets the jth polytope along a codimension 2 face, we add the quantity L ik sinh ϕ ik , which is symmetric in its arguments. Hence the gradient of A is symmetric.

Let us consider the simplest case of Γ-convex polyhedra in the Minkowski plane, with only one support number h R. Then by ( 22) covolphq is equal to h 2 times a positive number, in particular it is a strictly convex function. This is always true.

Theorem 4.5. The Hessian of covol : R n Ñ R is positive definite.

Recall that we are looking at the covolume on a space of support vectors, and not on a space of polyhedra (the sum is not the same).

Proof. Due to (28) it suffices to study the Jacobian of A. The elements off the diagonal are non-positive due to (32). Note that the formula is also correct if E j i is empty. The diagonal terms (33) are positive, as any facet F i has an adjacent facet. As cosh x ¡ 1 for x 0, (33) and (32) lead to fApF i q fhpiq ¡ jIztiu § § § § fApF i q fhpjq § § § § ¡ 0 that means that the Jacobian is strictly diagonally dominant with positive diagonal entries, hence positive definite, see for example [Varga, 2000, 1.22].

Polyhedral Minkowski Theorem

We use a classical continuity method, although its Euclidean analog is more often proved using a variational method.

Theorem 4.6 (Minkowski Theorem). Let Γ be a Fuchsian group, R pη 1 , . . . , η n q be a set of pairwise non collinear unit future time-like vectors of the Minkowski space contained in a fundamental domain of Γ, and let pf 1 , . . . , f n q be positive real numbers.

There exists a unique Γ-convex polyhedron with inward unit normals η i such that the facet orthogonal to η i has area f i .

Theorem 4.6 is equivalent to say that the map Φ from PpΓ, Rq to pR q n which associates to each ph 1 , . . . , h n q PpΓ, Rq the facet areas pApF 1 q, . . . , ApF n qq is a bijection. By Lemma 4.4, Theorem 4.5 and local inverse theorem, Φ is locally invertible. So Φ is a local homeomorphism by the invariance of domain theorem. Lemma 4.7 below says that Φ is proper. As pR q n is connected, it follows that Φ is surjective, hence a covering map. But the target space pR q n is simply connected and PpΓ, Rq is connected (Lemma 4.3), so Φ is a homeomorphism, in particular bijective, and Theorem 4.6 is proved.

Lemma 4.7. The map Φ is proper: Let pa α q αN be a converging sequence of pR q n such that for all α, there exists h α ph α p1q, . . . , h α pnqq PpΓ, Rq with Φph α q a α . Then a subsequence of ph α q α converges in PpΓ, Rq. Proof. Let α N and suppose that h α piq is the largest component of h α . For any support number h α pjq, j ΓI, of a facet adjacent to the one supported by h α piq, as h α piq ¥ h α pjq, (22) gives:

h α i j h α piq cosh ϕ i j ¡ h α pjq sinh ϕ i j ¥ h α piq cosh ϕ i j ¡ 1 sinh ϕ i j .
As Γ acts cocompactly on H d , for any j ΓI, ϕ i j is bounded from below by a positive constant. Moreover the function x Þ Ñ cosh x¡1 sinh x is increasing, then there exists a positive number λ i , depending only on i, such that

h α i j ¥ h α piqλ i .
As the sequence of areas of the facets is supposed to converge, there exists positive numbers A i and A ¡ i such that A i ¥ ApF α i q ¥ A ¡ i , where ApF α i q is the area of the facet F α i supported by h α piq. A ¡ i ), the isoperimetric inequality gives [Burago and Zalgaller, 1988, 10.1]

j L α i j ¥ Per α i ¥ Per i ,
where the sum is on the facets adjacent to F α i and L α i j is the pd ¡ 1q volume of the codimension 2 face between F α i and F α j . We get

A i ¥ ApF α i q 1 d j h α i j L α i j ¥ h α piqλ i 1 d j L α i j ¥ h α piq λ i Per i d .
As h α piq is the largest component of h α , all the support numbers are bounded from above by a constant which does not depend on α. Moreover each component of h α is positive, hence all the components of the elements of the sequence ph α q α are bounded from above and below, so there exists a subsequence ph ϕpαq q ϕpαq converging to php1q, . . . , hpnqq, where hpiq is a non-negative number. Suppose that the limit of ph ϕpαq piqq ϕpαq is zero. Let h ϕpαq pjq be the support number of a facet adjacent to

F ϕpαq i
. If ϕpαq is sufficiently large, h ϕpαq pjq is arbitrary close to hp jq, which is a non-negative number, and h ϕpαq piq is arbitrary close to 0. By ( 22), h α i j is a non-positive number. So all the support numbers of F

Mixed face area and mixed-covolume

Let us recall some basic facts about convex polytopes in Euclidean space (with non empty interior). A convex polytope of R d is simple if each vertex is contained in exactly d facets. Each face of a simple convex polytope is a simple convex polytope. The normal fan of a convex polytope is the decomposition of R d by convex cones defined by the outward unit normals to the facets of the polytope (each cone corresponds to one vertex). Two convex polytopes are strongly isomorphic if they have the same normal fan. The Minkowski sum of two strongly isomorphic simple polytopes is a simple polytope strongly isomorphic to the previous ones. Moreover the support vector of the Minkowski sum is the sum of the support vectors.

Let Q be a simple convex polytope in R d with n facets. The set of convex polytopes of R d strongly isomorphic to Q is a convex open cone in R n . The Euclidean volume V E is a polynomial of degree d on this set, and its polarization V E p¤, . . . , ¤q is the mixed-volume. The coefficients of the volume depend on the combinatorics, it's why we have to restrict ourselves to simple strongly isomorphic polytopes. The following result is an equivalent formulation of the Alexandrov-Fenchel inequality. Theorem 4.8 ([Alexandrov, 1996, Schneider, 1993]). Let Q, Q 3 , . . . , Q d be strongly isomorphic simple convex polytopes of R d with n facets and Z R n . Then

V E pZ, Q, Q 3 , . . . , Q d q 0 ñ V E pZ, Z, Q 3 , . . . , Q d q ¤ 0
and equality holds if and only if Z is the support vector of a point.

We identify a support hyperplane of an element of PpΓ, Rq with the Euclidean space R d by performing a translation along the ray from the origin orthogonal to the hyperplane. In this way we consider all facets of elements of PpΓ, Rq lying in parallel hyperplanes as convex polytopes in the same Euclidean space R d .

The definition of strong isomorphy and simplicity extend to Γ-convex polyhedra, considering them as polyhedral hypersurface in the ambient vector space. Note that the simplest examples of Euclidean convex polytopes, the simplices, are simple, but the simplest examples of Γ-convex polyhedra, those defined by only one orbit, are not simple (if d ¡ 1). Let us formalize the definition of strong isomorphy. The normal cone NpPq of a convex Γ-polyhedron P is the decomposition of F by convex cones defined by the inward normals to the facets of P. It is the minimal decomposition of F such that the extended support functions of P is the restriction of a linear form on each part. If the normal fan NpQq subdivides NpPq, then we write NpQq ¡ NpPq. Note that NpP Qq ¡ NpPq. Two convex Γ-polyhedron P and Q are strongly isomorphic if NpPq NpQq. If P is simple, we denote by rPs the subset of PpΓ, Rq made of polyhedra strongly isomorphic to P. Lemma 4.9. All elements of rPs are simple and rPs is an open convex cone of R n . Proof. The fact that all elements of rPs are simple and that rPs is open are classical, see for example [Alexandrov, 1937].

The only difference with the Euclidean convex polytopes case is that, around a vertex, two facets can belong to the same orbit for the action of Γ, hence when one wants to slightly move a facet adjacent to a vertex, one actually moves two (or more) facets. But this does not break the simplicity, nor the strong isomorphy class.

Moreover rPs is a convex cone as the sum of two functions piecewise linear on the same decomposition of F gives a piecewise linear function on the same decomposition.

Suppose that P is simple, has n facets (in a fundamental domain), and let h 1 , . . . , h d 1 rPs (support vectors of polyhedra strongly isomorphic to P). Let us denote by F k piq the ith facet of the polyhedron with support vector h k , and let hpF k piqq be its support vector (F k piq is seen as a convex polytope in R d ). The entries of hpF k piqq have the form (22) so the map h k Þ Ñ hpF k piqq is linear. This map can be defined formally for all Z R n using (22). The mixed face area Aph 2 , . . . , h d 1 q is the vector formed by the entries V E phpF 2 piqq, . . . , hpF d 1 piqqq, i 1, . . . , n. Together with (26), this implies that covol is a pd 1q-homogeneous polynomial, and we call mixed-covolume its polarization covolp¤, . . . , ¤q. Note that covol is C V on rPs. Lemma 4.10. We have the following equalities, for X i R n .

(i) D d¡1 5.2 Mixed covolume and standard inequalities Lemma 5.4. The covolume on KpΓq is a homogeneous polynomial of degree pd 1q. Its polar form is the mixed-covolume covolp¤, . . . , ¤q, a continuous non-negative symmetric map on pKpΓqq d 1 such that covolpK, . . . , Kq covolpKq.

Moreover if we restrict to a space of strongly isomorphic simple Γ-convex polyhedra, or to the space of C V Γ-convex bodies, then covolp¤, . . . , ¤q is the same map as the one previously considered.

Proof. Let us define covolpK 1 , . . . ,

K d 1 q 1 pd 1q! d 1 i1 p¡1q d 1 k i1 ¤¤¤ i d 1 covolpK i 1 ¤ ¤ ¤ K i d 1 q (35)
which is a symmetric map. From the continuity of the covolume and of the Minkowski addition, it is a continuous map. In the case when K i are strongly isomorphic simple polyhedra, the right-hand side of ( 35) to the mixed-covolumes previously introduced [Schneider, 1993, 5.1.3] (we also could have used another polarization formula [Hörmander, 2007, (A.5)]). Let us consider a sequence of strongly isomorphic simple Γ-convex polyhedra P 1 pkq, . . . , P d 1 pkq converging to K 1 , . . . , K d 1 (Lemma 5.2). From the definition of the mixed-covolume we have

covolpλ 1 P 1 pkq ¤ ¤ ¤ λ d 1 P d 1 pkqq d 1 i1 ,...,i d 1 1 λ i 1 ¤ ¤ ¤ λ i d 1 covolpPpkq i 1 , . . . , Ppkq i d 1 q
and by continuity, passing to the limit,

covolpλ 1 K 1 ¤ ¤ ¤ λ d 1 K d 1 q d 1 i1 ,...,i d 1 1 λ i 1 ¤ ¤ ¤ λ i d 1 covolpK i 1 , . . . , K i d 1 q
so the covolume is a polynomial, and covolp¤, . . . , ¤q introduced at the beginning of the proof is its polarization.

It is non-negative due to Corollary 4.11.

In the case of C 2 Γ-convex bodies, both notions of mixed-covolume satisfy (35).

Theorem 5.5. Let K i KpΓq and 0 t 1. We have the following inequalities.

Reversed Alexandrov-Fenchel inequality:

covolpK 1 , K 2 , K 3 , . . . , K d 1 q 2 ¤ covolpK 1 , K 1 , K 3 , . . . , K d 1 qcovolpK 2 , K 2 , K 3 , . . . , K d 1 q
First reversed Minkowski inequality:

covolpK 1 , K 2 , . . . , K 2 q d 1 ¤ covolpK 2 q d covolpK 1 q
Second or quadratic reversed Minkowski inequality:

covolpK 1 , K 2 , . . . , K 2 q 2 ¤ covolpK 2 qcovolpK 1 , K 1 , K 2 , . . . , K 2 q Reversed Brunn-Minkowski inequality: covolpp1 ¡ tqK 1 tK 2 q 1 d 1 ¤ p1 ¡ tqcovolpK 1 q 1 d 1 tcovolpK 2 q 1 d 1
Reversed linearized first Minkowski inequality:

pd 1qcovolpK 1 , K 2 , . . . , K 2 q ¤ dcovolpK 2 q covolpK 1 q
If all the K i are C V or strongly isomorphic simple polyhedra, then equality holds in reversed Alexandrov- Fenchel and second reversed Minkowski inequalities if and only if K 1 and K 2 are homothetic.

In the classical case of Euclidean convex bodies, the linearized first Minkowski inequality is valid only on particular subsets of the space of convex bodies, see [Schneider, 1993, (6.7.11)].

Proof. Let P 1 pkq, . . . , P d 1 pkq be a sequence of simple strongly isomorphic Γ-convex polyhedra converging to K 1 , . . . , K d 1 (Lemma 5.2). Applying Cauchy-Schwarz inequality to the inner product covolp¤, ¤, P 3 pkq, . . . , P d 1 pkqq (Theorem 4.13) at pP 1 pkq, P 2 pkqq and passing to the limit gives reversed Alexandrov-Fenchel inequality. Equalities cases follow from Theorem 3.11 and 4.13. The second reversed Minkowski inequality and its equality case follows from Alexandrov-Fenchel inequality.

As the covolume is convex (Theorem 5.3), for K 1 and K 2 of unit covolume, for t r0, 1s we get covolpp1 ¡ tqK 1 tK 2 q ¤ 1.

Taking K i K i {covolpK i q 1 d 1 and

t tcovolpK 2 q 1 d 1 p1 ¡ tqcovolpK 1 q 1 d 1 tcovolpK 2 q 1 d 1
leads to the reversed Brunn-Minkowski inequality.

As covolp¤q is convex, the map f pλq covolpp1 ¡ λqK 1 λK 2 q ¡ p1 ¡ λqcovolpK 1 q ¡ λcovolpK 2 q, 0 ¤ λ ¤ 1, is convex. As f p0q f p1q 0, we have f I p0q ¤ 0, that is the reversed linearized first Minkowski inequality.

(Remember that covolpp1 ¡ λqK 1 λK 2 q p1 ¡ λq d 1 covolpK 1 q pd 1qp1 ¡ λq d λcovolpK 1 , . . . , K 1 , K 2 q λ 2 r. . .s.q Reversed Brunn-Minkowski says that the map covolp¤q 1 d 1 is convex. Doing the same as above with the convex map gpλq covolpp1 ¡ λqK 1 λK 2 q 1 d 1 ¡ p1 ¡ λqcovolpK 1 q 1 d 1 ¡ λcovolpK 2 q 1 d 1 , 0 ¤ λ ¤ 1, leads to the first reversed Minkowski inequality.

The (Minkowski) area S pKq of a Γ-convex body K is pd 1qcovolpB, K, . . . , Kq. Note that it can be defined from the covolume:

S pKq lim εÑ0 covolpK εBq ¡ covolpKq ε .

The following inequality says that, among Γ-convex bodies of area 1, B has smaller covolume, or equivalently that among Γ-convex bodies of covolume 1, B has larger area. Lemma 5.7. If K is a C V Γ-convex body, then S pKq is the volume of the Riemannian manifold fK{Γ.

If K is a Γ-convex polyhedron, then S pKq is the total face area of K (the sum of the area of the facets of K in a fundamental domain).

In particular S pBq is the volume of the compact hyperbolic manifold H d {Γ.

Proof. The C 2 case follows from the formulas in Section 3, because B is a C 2 convex body.

Let K be polyhedral. Let pP k q k be a sequence of polyhedra converging to B and such that all the support numbers of P k are equal to 1 (i.e. all facets are tangent to H d ). Up to add facets, we can construct P k such that NpP k q ¡ NpKq and P k is simple. Let α be a small positive number. The polyhedron K αP k is strongly isomorphic to P k . It follows from formulas of Section 4 than pd 1qcovolpP k , K αP k , . . . , K αP k q is equal to the total face area of K αP k . By continuity of the mixed-covolume, pd 1qcovolpP k , K αP k , . . . , K αP k q converges to pd 1qcovolpP k , K, . . . , Kq when α goes to 0.

We associate to K a support vector hpKq whose entries are support numbers of facets of K, but also to support hyperplanes of K parallel to facets of P k . We also consider "false faces" of larger codimension, such that the resulting normal fan is the same as the one of P k . This is possible as the normal fan of P k is finer than the one of K. The support numbers of the false faces can be computed using ( 21) and ( 22) (i.e. K is seen as an element of the closure of rP k s). In particular hpK αP k q hpKq αhpP k q and as the map s giving the support numbers of a facet in terms of the support numbers of the polyhedron is linear, the area of this facet is V E psphpKqq αsphpP k qqq. By continuity of the Euclidean volume, when α goes to 0 this area goes to the area of the facet of K (it is 0 if the facet was a "false facet" of K). Hence pd 1qcovolpP k , K, . . . , Kq is equal to the total face area of K, and on the other hand it goes to S pKq when k goes to infinity.

Let us end with an example. Let K be a polyhedral Γ-convex body with support numbers equal to 1. In this case S pKq pd 1qcovolpKq, and as S pBq pd 1qcovolpBq, the isoperimetric inequality becomes S pKq S pBq ¤ 1. Let d 2 and Γ be the Fuchsian group which has a regular octagon as a fundamental domain in the Klein model of H 2 . Then by the Gauss-Bonnet theorem S pBq 4π. The total face area of K is the area of only one facet, which is eight times the area of a Euclidean triangle of height h I cosh ϕ¡1 sinh ϕ and with edge length two times h I 1¡cos π{4 sin π{4 (see ( 21) and ( 22)). ϕ is the distance between a point of H 2 and its image by a generator of Γ, and cosh ϕ 2 2 c 2 (compare Example C p. 95 in [Katok, 1992] with Lemma 12.1.2 in [START_REF] Maclachlan | The arithmetic of hyperbolic 3-manifolds[END_REF]). By a direct computation the isoperimetric inequality becomes 0, 27 13 ¡ 9 c 2 ¤ π 2 1, 57.

Remarks on equality cases and general Minkowski theorem Brunn-Minkowski inequality for non-degenerated (convex) bodies in the Euclidean space comes with a description of the equality case. Namely, the equality occurs for a t if and only if the bodies are homothetic (the part "if" is trivial). At a first sigh it is not possible to adapt the standard proof of the equality case to the Fuchsian case, as it heavily lies on translations [START_REF] Bonnesen | Theory of convex bodies[END_REF], Schneider, 1993, Alexandrov, 2005].

If such a result was known, it should imply, in a way formally equivalent to the classical one, the characterization of the equality case in the reversed first Minkowski inequality, as well as the uniqueness part in the Minkowski theorem and the equality case in the isoperimetric equality (see below).

The Minkowski problem in the classical case is to find a convex body having a prescribed measure as "area measure" (see notes of Section 5.1 in [Schneider, 1993]). It can be solved by approximation (by C 2 or polyhedral convex bodies), see [Schneider, 1993], or by a variational argument using the volume, see [Alexandrov, 1996]. Both methods require a compactness result, which is known as the Blaschke selection Theorem. Another classical question about Minkowski problem in the C 2 case, is to know the regularity of the hypersurface with respect to the regularity of the curvature function, see the survey [START_REF] Trudinger | The Monge-Ampère equation and its geometric applications[END_REF]. All those questions can be transposed in the setting of Fuchsian convex bodies.

Figure 1 :

 1 Figure 1: Recovering the convex body from its support function in the Minkowski space.

  ¡ d fK,where η is the unit outward normal of fK{Γ in PpKq{Γ (hence it corresponds in the universal cover to the unit inward normal of K). If X is the position vector in F we getpd 1qcovolpKq ¡ » fK{Γ h ¥ N d fKwith h the support function of K and N the Gauss map.

Figure 2 :

 2 Figure 2: A piece of a Γ-convex polyhedron in d 2 seen from the bottom. It is made with the orbit of p0, 0, 1q for the Fuchsian group having a regular octagon as fundamental domain in H 2 .

Figure 3 :

 3 Figure 3: Supports numbers of a Γ-convex polyhedron.

  Lemma 4.3. The set PpΓ, Rq is a non-empty open convex cone of R n .

  ) is true. As made of volumes of convex polytopes of R d , A is homogeneous of degree d so by Euler homogeneous theorem D h Aphq dAphq. Using this in (29) with Y h gives d d

  , hence the d volume of F ϕpαq i is non-positive, that is impossible. It follows easily that ph ϕpαq piqq ϕpαq converges in PpΓ, Rq.

  Corollary 5.6 (Isoperimetric inequality). Let K be a Γ-convex body. Then It follows from the first reversed Minkowski with K 1 B, K 2 K, divided by S pBq d 1 , with pd 1qcovolpBq S pBq.

  There exists t such that Γx H d t either be a horosphere by Sublemma 2.3. Hence H must be space-like, that gives (ii)(b). The fact that all elements of Γx belong to H d t implies that K is in the future side of its support planes, hence (ii)(c). As the action of Γ on H d is cocompact, it is well-known that the orbit Γx is discrete and that the Dirichlet regions for Γx D a pΓq tp H d |dpa, pq ¤ dpγa, pq, dγ ΓztIduu, a Γx

	t , and all elements of t can't be a totally geodesic hyperplane (of H d t . Clearly H H d t on H d Γx must be on one side of H H d t ), and it can'(1)

Sublemma 2.3. Let Γ be a group of isometries acting cocompactly on the hyperbolic space H d . For any x H d , the orbit Γx meets the interior of any horoball.

Proof.

  ].A future time-like vector η orthogonal to a support plane at x of a future convex set A is called an inward normal of A at x. This means that dy A, y ¡ x and η are two future time-like vectors at the point x, then by Sublemma 2.5, dy A, xη, y ¡ xy ¡ ¤ 0, i.e. xη, yy ¡ ¤ xη, xy ¡ or equivalently the sup on all y A of xη, yy ¡ is attained at x. Notice that the set ty R d 1 |xy, ηy ¡ xx, ηy ¡ u

	Sublemma 2.5.	(i) If x and y are nonzero non space-like vectors in R d 1 , both past or future, then xx, yy ¡ ¤ 0
	with equality if and only if x and y are linearly dependent light-like vectors.
	(ii) If x and y are nonzero non space-like vectors in R d 1 , both past (resp. future), then the vector x y is past
	(resp. future) non space-like. Moreover x y is light-like if and only if x and y are linearly dependent
	light-like vectors.

  Sublemma 2.11. Let H be a sublinear function on F with finite values. Let us extend it as a convex function on R d 1 by giving the value V outside F . Let H be the lower semi-continuous hull of H: Hpxq liminf xÑy Hpyq.

	• If η is future light-like, then Hpηq 0. As H is lower semi-continuous (as supremum of a family of continuous functions) and as H V outside of the future cone, this follows from Sublemma 2.11. • By definition, Hp0q 0.
	That H is a Γ invariant sublinear function follows easily.

  body, h its support function and H its extended support function (h is the restriction of H to H d ). For any η H d we have hpηq xN ¡1 pηq, ηy ¡ , (10) and for η F , introducing the 0-homogeneous extension Ñ¡1 of N ¡1 we obtain D η HpXq x Ñ¡1 pηq, Xy ¡ xD η Ñ¡1 pXq, ηy ¡ , but D η Ñ¡1 pXq belongs to the support hyperplane of K with inward normal η so D η HpXq x Ñ¡1 pηq, Xy ¡ . HpX, Yq xB ¡1 pXq, Yy ¡ , in particular H is C 2 , so h is C 2 and (9) is known. As h is Γ-invariant, we get a function of C 2 pΓq. Now let h C 2 pΓq. We also denote by h the Γ-invariant map on H d which projects on h, and by H the 1-homogeneous extension of h to F . The 1-homogeneity and (9) imply that H is convex (in the hyperbolic coordinates, row and column of the Hessian of H corresponding to the radial direction r are zero), hence negative sublinear Γ-invariant, so it is the support function of a Γ-convex body K by Lemma 2.12. As h is C 2 , we get a map G from H d to fK R d 1 which is C 1 , and regular from (

	Hence D 2 η

  . All the elements of Γη i belong to H d t i , on which Γ acts cocompactly. Up to a homothety, it is more suitable to consider that H d t i is H d 1 H d . Let a Γη i and D a pΓq be the Dirichlet region (see (1)). Recall that D a pΓq are convex compact polyhedra in H d , and that the set of the Dirichlet regions D a , for all a Γη i , is a locally finite tessellation of H d . Using (20), the Dirichlet region can be written D a pΓq tp H d |xa, py ¡ ¥ xγa, py ¡ , dγ ΓztIduu. Let a 1 , a 2 Γη i such that D a 1 pΓq and D a 2 pΓq have a common facet. This facet is contained in the intersection of H d with the hyperplane tp R d 1 |xa 1 , py ¡ xa 2 , py ¡ u,

	and this hyperplane also contains a u 1 a u 2 by (19). It follows that vertices of P i (codimension pd 1q faces)

  If Per α i (resp. Per i )is the Euclidean pd ¡ 1q volume of the hypersphere bounding the ball with Euclidean d volume ApF α i q (resp.

X 2 ApX 3 , . . . , X d 1 q d!ApX 2 , . . . , X d 1 q, (ii) D X 1 covolpX 2 q pd 1qcovolpX 2 , X 1 , . . . , X 1 q, (iii) D 2 X 1 covolpX 2 , X 3 q pd 1qdcovolpX 2 , X 3 , X 1 , . . . , X 1 q, (iv) D d X 1 covolpX 2 , . . . , X d 1 q pd 1q!covolpX 1 , . . . , X d 1 q, (v) covolpX 1 , . . . , X d 1 q 1 d 1 d X 1 , ApX 2 , . . . , X d 1 q h .Proof. The proof is analogous to the one of Lemma 3.9.Corollary 4.11. For h i rPs, covolph 1 , . . . , h d 1 q is non-negative.

Lemma 3.9. We have the following equalities, for X i C V pΓq.

(i) D d¡1 X 2 κ ¡1 pX 3 , . . . , X d 1 q d!κ ¡1 pX 2 , . . . , X d 1 q, (ii) D X 1 covolpX 2 q pd 1qcovolpX 2 , X 1 , . . . , X 1 q, (iii) D 2 X 1 covolpX 2 , X 3 q pd 1qdcovolpX 2 , X 3 , X 1 , . . . , X 1 q, (iv) D d X 1 covolpX 2 , . . . , X d 1 q pd 1q!covolpX 1 , . . . , X d 1 q, (v) covolpX 1 , . . . , X d 1 q ¡ 1 d 1 pX 1 , κ ¡1 pX 2 , . . . , X d 1 qq.

Proof. (i) and (iv) are proved by induction on the order of the derivative, using the definition of directional derivative and the expansion of the multilinear forms. (ii) and (iii) are obtained by the way. (v) follows from ( 14), (i) and (iv).

Corollary 3.10. For h i C V pΓq, covolph 1 , . . . , h d 1 q is positive.

Proof. As h i C V pΓq, h P i is positive definite, hence κ ¡1 ph 2 , . . . , h d 1 q ¡ 0 [Alexandrov, 1996, (5) p. 122].

The result follows from (v) because h 1 0.

Due to (iii) of the preceding lemma, the following result implies Theorem 3.5.

Theorem 3.11. For any h 1 , . . . , h d¡1 in C V pΓq, the symmetric bilinear form on pC V pΓqq 2 covolp¤, ¤, h 1 , . . . , h d¡1 q is positive definite.

Proof. We use a continuity method. We consider the paths

where h 0 is the (quotient of the) support function of B d and we denote covol t p¤, ¤q : covolp¤, ¤, h 1 ptq, . . . , h d¡1 ptqq.

The result follows from the facts: (i) covol 0 is positive definite, (ii) if, for each t 0 r0, 1s, covol t 0 is positive definite, then covol t is positive definite for t near t 0 , (iii) if t n r0, 1s with t n Ñ t 0 and covol t n is positive definite, then covol t 0 is positive definite. (i) is Lemma 3.6. Let t 0 as in (ii). By Lemma 3.12, each κ ¡1 p¤, h 1 ptq, . . . , h d¡1 ptqq inherits standard properties of elliptic self-adjoint operators on compact manifolds (see for example [Nicolaescu, 2007]), and we can apply [Kato, 1995, Theorem 3.9 p. 392]: as the deformation of the operators is polynomial in t, the eigenvalues change analytically with t, for t near t 0 . In particular if t is sufficiently close to t 0 , the eigenvalues remain positive and (ii) holds.

Let t n be as in (iii). For any non zero X C V pΓq we have covol t n pX, Xq ¡ 0 with

As κ ¡1 is multilinear and as t n 1, it is easy to see that the function in the integrand above is bounded by a function (of the kind X °|κ ¡1 pX, ¦, . . . , ¦q| where each ¦ is h 0 or a h i ) which does not depend on n and is continuous on the compact M. By Lebesgue's dominated convergence theorem, covol t 0 pX, Xq ¥ 0, and by Lemma 3.13 covol t 0 pX, Xq ¡ 0, and (iii) is proved. Lemma 3.12. For any h 1 , . . . , h d¡1 in C V pΓq, the operator κ ¡1 p¤, h 1 , . . . , h d¡1 q is formally self-adjoint linear second order elliptic.

Proof. As h i are support vectors of strongly isomorphic simple polyhedra, the entries of Aph 2 , . . . , h d 1 q are mixed-volume of simple strongly isomorphic Euclidean convex polytopes, hence are non-negative (see Theorem 5.1.6 in [Schneider, 1993]). The result follows from (v) because the entries of h 1 are positive.

Lemma 4.12. For any h 1 , . . . , h d¡1 rPs, the symmetric bilinear form covolp¤, ¤, h 1 , . . . , h d¡1 q has trivial kernel.

Proof. The analog of the proof of Lemma 3.13, using Theorem 4.8 instead of Theorem 3.8, gives that in each support hyperplane, the "support vectors" of Z (formally given by ( 22)) are the ones of a point of R d . Let us denote by Z i the support vector of Z in the hyperplane with normal η i .

If ε is sufficiently small then h 1 εZ is the support vector of a Γ-convex polyhedron P ε 1 strongly isomorphic to P 1 , the one with support vector h 1 . Moreover the support numbers of the ith facet F i of P ε 1 are the sum of the support numbers of the facet F 1 i of P 1 with the coefficients of εZ i . As Z i is the support vector of a point in R d , F i is obtained form F 1 i by a translation. It follows that each facet of P ε 1 is obtained by a translation of the corresponding facet of P 1 , hence P ε 1 is a translate of h 1 (the translations of each facet have to coincide on each codimension 2 face). As h 1 εZ is supposed to be a Γ-convex polyhedron for ε sufficiently small, and as the translation of a Γ-convex polyhedron is not a Γ-convex polyhedron, it follows that Z 0. Theorem 4.13. For any h 1 , . . . , h d¡1 rPs, the symmetric bilinear form covolp¤, ¤, h 1 , . . . , h d¡1 q is positive definite.

Proof. The proof is analogous to the one of Theorem 3.11.

Remark on spherical polyhedra

The sets of strongly isomorphic simple Γ-convex polyhedra form convex cones in vector spaces (Lemma 4.9). The mixed-covolume allow to endow these vector spaces with an inner product. Hence, if we restrict to polyhedra of covolume 1, those sets are isometric to convex spherical polyhedra. For d 1 we get simplices named orthoschemes [START_REF] Fillastre ; Fillastre | Polygons of the Lorentzian plane and spherical polyhedra[END_REF]. In d 2, if we look at the metric induced on the boundary of the Fuchsian polyhedra, we get spherical metrics on subsets of the spaces of flat metrics with cone-singularities of negative curvature on the compact surfaces of genus ¡ 1. It could be interesting to investigate the shape of these subsets.

5 General case

Convexity of the covolume

Hausdorff metric Recall that KpΓq is the set of Γ-convex bodies for a given Γ. For K, K I we define the Hausdorff metric by dpK, K I q mintλ ¥ 0|K I λB K, K λB K I u.

It is not hard to check that this is a distance and that Minkowski sum and multiplication by a positive scalar are continuous for this distance. If we identify Γ-convex bodies with their support functions, then KpΓq is isometric to a convex cone in C 0 pH d {Γq endowed with the maximum norm, i.e.:

dpK, K I q sup

The proofs is easy and formally the same as in the Euclidean case [Schneider, 1993, 1.8.11].

Lemma 5.1. The covolume is a continuous function.

Proof. Let K be in KpΓq with support function h. For a given ε ¡ 0, choose λ ¡ 1 such that pλ d 1 ¡1qλ d 1 covolpKq ε.

Let ρ 0 such that h ¡ ρ, and let α ¡ 0 be the minimum of h ¡ρ. Let α minpα, p1¡λqρq ¡ 0. In particular, ρ ¤ h ¡ α.

(34)

Finally, let K with support function h be such that dpK, Kq α. In particular, h ¡ α h, that, inserted in (34),

gives that ρ h. This and the definition of α give h ¤ h α ¤ h p1 ¡ λqρ ¤ h p1 ¡ λqh, i.e. λh ¤ h, i.e. λK K, in particular covolpKq ¤ λ d 1 covolpKq. In a similar way we get covolpKq ¤ λ d 1 covolpKq. This allows to write

The general results are based on polyhedral approximation.

Lemma 5.2. Let K 1 , . . . , K p KpΓq. There exists a sequence pP 1 k , . . . , P p k q k of strongly isomorphic simple Γ-convex polyhedra converging to pK 1 , . . . , K p q.

Proof. First, any Γ-convex body K is arbitrarily close to a Γ-convex polyhedron Q. Consider a finite number of points on K and let Q be the polyhedron made by the hyperplanes orthogonal to the orbits of these points, and passing through these points. We get K Q. For any ε ¡ 0, if Q εB is not included in K then add facets to Q. The process ends by cocompactness.

Let Q i be a Γ-convex polyhedron arbitrary close to K i , and let P be the

Let us suppose that around a vertex x of P, two facets belong to the same orbit for the action of Γ. We perform a little translation in direction of P of a support hyperplane at x, which is not a support hyperplane of a face containing x. A new facet appears, the vertex x disappears, and the two facets in the same orbit share one less vertex. Repeating this operation a finite number of times, we get a polyhedron P I with NpP I q ¡ NpPq and such that around each vertex, no facets belong to the same orbit. If P I is not simple, there exists a vertex x of P I such that more than d 1 facets meet at this vertex. We perform a small little parallel move of one of this facets. In this case the number of facets meeting at the vertex x I corresponding to x decreases, and new vertices can appear, but the number of facets meeting at each of those vertices is strictly less than the number of facets meeting at x. If the move is sufficiently small, the number of facets meeting at the other vertices is not greater than it was on P I . Repeating this operation a finite number of times leads to the simple polyhedra P P , and NpP P q ¡ NpP I q.

Now we define P i Q i αP P , with α ¡ 0 sufficiently small such that P i remains close to Q i and hence close to K i . By definition of P, NpPq ¡ NpQ i q and finally NpP P q ¡ NpQ i q hence NpP i q NpP P q: all the P i are strongly isomorphic to P P , which is simple.

Theorem 5.3. The covolume is a convex function on the space of Γ-convex bodies: for any K 1 , K 2 KpΓq, dt r0, 1s, covolpp1 ¡ tqK 1 tK 2 q ¤ tcovolpK 1 q p1 ¡ tqcovolpK 2 q.

Proof. By Lemma 5.2, there exist strongly isomorphic simple Γ-convex polyhedra P 1 and P 2 arbitrary close to respectively K 1 and K 2 . As for simple strongly isomorphic Γ-convex polyhedra, the addition of support vectors is the same as Minkowski addition, Theorem 4.5 gives that covolpp1 ¡ tqP 1 tP 2 q ¤ tcovolpP 1 q p1 ¡ tqcovolpP 2 q and the theorem follows by continuity of the covolume.


Ahmed Boujraf

Driss Sbibih

Christophe Léger

Rachid Harba

A SPLINE QUASI-INTERPOLANT FOR FITTING 3D DATA ON THE SPHERE AND APPLICATIONS

Keywords: B-spline, Quasi-interpolant, Sphere-like surface, Medical imaging

, the authors have approached the sphere-like surfaces using the tensor product of an algebraic cubic spline quasiinterpolant with a 2π-periodic Uniform Algebraic Trigonometric B-splines (UAT B-splines) of order four. In this paper, we improve the results given in [1], by introducing a new quasi-interpolant based on the tensor product of an algebraic cubic spline quasi-interpolant with a periodic cubic spline quasi-interpolant, obtained by the periodization of an algebraic cubic spline quasi-interpolant. Our approach allows us to obtain an approximating surface which is of class C 2 and with an approximation order O(h 4 ). We show that this method is particularly well designed to render 3D closed surfaces, and it has been successfully applied to reconstruct human organs such as the left ventricle of the heart.

INTRODUCTION

Let S be a sphere-like surface, i.e. a surface of R 3 which is topologically equivalent to a sphere. Assume that we have a set of scattered points P 1 , . . . , P d located on S, along with real numbers r 1 , . . . , r d associated with each of these points. As in practice these numbers are derived, in general, from experiments and are almost always noisy, we are interested in this paper to find a function F defined on S which approximates them in the sense that

F (P i ) r i , 1 ≤ i ≤ d. ( 1
)
Data fitting problems where the underlying domain is a sphere-like surface arise in many areas, including e.g. geophysics and meteorology where S is taken as a model of earth (see [START_REF] Fasshauer | Scattered Data Fitting on the Sphere[END_REF]), and in medical modeling where S may be the surface of a human organ like heart, lung, bladder, kidney, etc. In most of these applications, the function F is constructed so that its associated closed surface S F = {F (s)s, s ∈ S} is at least continuous. In some cases, the C 1 or C 2 continuity is required.

Several methods have been proposed in the literature in the past fifteen years for fitting scattered data on sphere-like surfaces. For a survey of these methods on the sphere see [START_REF] Fasshauer | Scattered Data Fitting on the Sphere[END_REF] and references therein. Some other methods based on specific techniques have been developed in [START_REF] Goodman | Closed surfaces defined from biquadratic splines[END_REF], [START_REF] Lawson | C 1 surface interpolation for scattered data on a sphere[END_REF], [START_REF] Liu | Hybrid Bézier patches on sphere-like surfaces[END_REF], [START_REF] Neamtu | On the approximation order of splines on the spherical triangulations[END_REF], [START_REF] Feng | Closed smooth surface defined from cubic triangular splines[END_REF] and [START_REF] Schumaker | Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines[END_REF].

Among the methods developed for solving Problem (1) and based on quasi-interpolants, we encounter tensor spline methods. The principle of these methods consists in converting Problem (1) to one defined on a rectangle. More specifically and without loss of generality, when the surface S is the unit sphere, it can be identified with the rectangle

H = [-π 2 , π 2 ] × [0, 2π
] by the mapping χ defined by

χ : H -→ S (θ, φ) -→   cos(θ)cos(φ) cos(θ)sin(φ) sin(θ)   .
The representation f of F in polar coordinates, defined on H by f = F oχ, is identical to that of F , i.e. S f = S F = {f (θ, φ)χ(θ, φ), (θ, φ) ∈ H}. However, the smoothness properties of f are not equivalent to those of its corresponding closed surface S f . More specifically, the surface S f = S F is of class C p , p = 0, 1, 2, on S if and only if f ∈ C p and satisfies the following (2p + 2) conditions :

f (θ, 0) = f (θ, 2π), (C 1 ) f (± π 2 , φ) = c ± , (C 2 ) ∂f ∂φ (θ, 0) = ∂f ∂φ (θ, 2π), (C 3 ) ∂f ∂θ (± π 2 , φ) = a ± cos(φ) + b ± sin(φ), (C 4 ) ∂ 2 f ∂φ 2 (θ, 0) = ∂ 2 f ∂φ 2 (θ, 2π), (C 5 ) ∂ 2 f ∂θ 2 (± π 2 , φ) = d ± + e ± cos(2φ) + f ± sin(2φ), (C 6 )
with a ± , b ± , c ± , d ± , e ± and f ± are real constants (see [START_REF] Gmelig Meyling | B-spline ap proximation of a closed surface[END_REF]). Now, if we set

F p = {f ∈ C p (H) : conditions (C 1 ) - (C 2p+2 ) hold}, then the problem of finding F such that S F is of class C p and satisfies F (P i ) r i , 1 ≤ i ≤ d, be- comes equivalent to finding f belonging to F p and satisfies f (θ i , φ i ) r i , 1 ≤ i ≤ d, where (θ i , φ i ) are the polar coor- dinates of P i , i.e. χ(θ i , φ i ) = P i .
Since the problem is now posed on a rectangular domain, it is natural to use a tensor spline method. Then, in this case, the quasi-interpolant emanating from such method for approximating f has the following form

f (θ, φ) = m i=1 n j=1 c i,j v i (θ) v j (φ) (2)
where

{v 1 , . . . , v m } (resp. { v 1 , . . . , v n }) is a linearly inde- pendent set of functions on [-π 2 , π 2 ] (resp. on [0, 2π]).
In Section 2, we give some preliminary results on cubic polynomial B-splines and we construct an associated quasiinterpolant of order four. In Section 3, we construct the periodic quasi-interpolant based on cubic polynomial B-splines of order four. Section 4, is devoted to the construction of a local quasi-interpolant on the sphere. Finally, in Section 5, we illustrate the performance of the method with some numerical tests (see Figure 2 and Figure 3) and applications to the left ventricle of the heart (see Figure 5).

QUASI-INTERPOLANT BASED ON CUBIC POLYNOMIAL B-SPLINES

In this section we construct a quasi-interpolant based on cubic polynomial B-splines which will be used in Section 4.

For I = [-π 2 , π 2 ] and a given positive integer n, let Θ n = {θ i } n+3
i=-3 , with mesh length h = π n , be a uniform partition of the interval I defined by   

θ i = -π 2 + ih for i = 0, ..., n θ -3 = θ -2 = θ -1 = -π 2 θ n+1 = θ n+2 = θ n+3 = π 2
The associated polynomial spline space of order 4 is defined by

S 4 (I, Θ n ) = {s ∈ C 2 : s |[θi,θi+1] ∈ P 3 },
where P 3 is the polynomial space of degree ≤ 3. The classical normalized cubique B-splines

B 4 i satisfy supp(B 4 i ) = [θ i , θ i+4 ] and B 4 i (θ) > 0, for θ i < θ < θ i+4 .
They form a partition of unity, i.e. i , i = 1, . . . , n + 3} forms a basis of S 4 (I, Θ n ). We now construct a local linear operator Q 1 which maps a given function f onto the cubic spline space S 4 (I, Θ n ) and which has an optimal approximation order. This operator is the C 2 cubic spline quasi-interpolant defined by

Q 1 f := n+3 i=1 µ i (f )B 4 i , (3)
where the coefficients µ i (f ) are defined as linear combinations of some values and derivatives of f on the set Θ n in order to have the exactness of the quasi-interpolant Q 1 on P 3 , i.e. Q 1 p = p, for all p ∈ P 3 . More specifically, these coefficients are defined as follows

µ 1 (f ) = f (- π 2 ), µ 2 (f ) = f (- π 2 ) + h 3 f (- π 2 ), µ 3 (f ) = f (- π 2 ) + hf (- π 2 ) + h 2 3 f (- π 2 ), µ i (f ) = 1 6 (-f (θ i-3 ) + 8f (θ i-2 ) -f (θ i-1 )), 4 ≤ i ≤ n, µ n+1 (f ) = f ( π 2 ) -hf ( π 2 ) + h 2 3 f ( π 2 ), µ n+2 (f ) = f ( π 2 ) - h 3 f ( π 2 ), µ n+3 (f ) = f ( π 2 ).
Using classical theorems of approximation, see for example [START_REF] Devore | Constructive Approximation[END_REF], we can easily prove that

Q 1 f -f ∞,I = O(h 4 ).

PERIODIC QUASI-INTERPOLANT BASED ON CUBIC POLYNOMIAL B-SPLINES

In this section we construct a periodic quasi-interpolant based on cubic polynomial B-splines. It is obtained by the periodization of the quasi-interpolant introduced in Section 2. More precisely, for J = [0, 2π] and a given positive integer m, let Φ m = {φ j } m+3 j=-3 , with mesh length k = 2π m , be a uniform partition of the interval J defined by   

φ j = jk for j = 0, ..., m φ -3 = φ m-3 , φ -2 = φ m-2 , φ -1 = φ m-1 φ m+3 = φ 3 , φ m+2 = φ 2 , φ m+1 = φ 1
Then, this quasi-interpolant, can be written in the form

Q 2 f := m+3 j=1 ν j (f )B 4 j , (4)
where

ν j (f ) = 1 6 (-f j-3 + 8f j-2 -f j-1 ), for 1 ≤ j ≤ m + 3,
and

f j = f (φ j ), ∀j.
It easy to see that for any periodic fonction f ∈ C 4 (J) of period 2π we have

Q 2 f -f ∞,J ≤ Ck 4 f (4) ∞,J ,
where C is a constant independent of m (see [START_REF] Sablonnière | Univariate spline quasi-interpolants and applications to numerical analysis[END_REF]).

QUASI-INTERPOLANT ON THE SPHERE

Now, we construct in this section a quasi-interpolant Q obtained by the tensor product of Q 1 and Q 2 defined in the preceding sections. This quasi-interpolant is given by

Qf (θ, φ) := n+3 i=1 m+3 j=1 µ i (ν j (f ))B 4 i (θ)B 4 j (φ) (5)
where

f ∈ F 2 = {s ∈ C 2 (H) : conditions (C 1 ) -(C 6 ) hold}.
It is obvious that this local linear operator Qf is of class C 2 on H and with an approximation ordre O(h 4 ).

NUMERICAL RESULTS AND APPLICATIONS TO MEDICAL IMAGING

Comparaison and application to synthetic data

To test the general performance of our method on synthetic data, we use the following function f defined explicitly on the rectangular domain H by

f (θ, φ) = 3 i=1 (g i (θ, φ)) -1/2 , where g i (θ, φ) = cos(θ)cos(φ) α i 2 + cos(θ)sin(φ) α i+1 2 + sin(θ) α i+2 2 with (α 1 , α 2 , α 3 , α 4 , α 5 ) = (5, 1, 2, 5, 1)
, (see Figure 1). It is easy to verify that f ∈ {s ∈ C 2 (H) : conditions (C i ) are satisfied, ∀i = 1, ..., 6} (see [START_REF] Dierckx | Algorithms for smoothing data on the sphere with tensor product splines[END_REF]).

We present in the following table a comparison on the error and the computation time corresponding to the quasiinterpolant Q and the quasi-interpolant Q studied in [START_REF] El Bachir Ameur | New Spline Quasi-Interpolant for Fitting 3D Data on the Sphere : Applications to Medical Imaging[END_REF] 1. The MSE and the computation time of the quasiinterpolant Q compared with the quasi-interpolant Q given in [START_REF] El Bachir Ameur | New Spline Quasi-Interpolant for Fitting 3D Data on the Sphere : Applications to Medical Imaging[END_REF].

Definition 1 The M SE between S f and S Qf was calculated as follow :

M SE = 1 n.m n i=1 m j=1 (Qf (θ i , φ j ) -f (θ i , φ j )) 2 ,
where n and m represent the mesh sizes.

In the following figures, we illustrate the graph of the closed surface associated with f (see Figure 1), and The graph of the closed surface associated with Qf for m = n = 8 and m = n = 16 (see Figure 2 and Figure 3). We notice that the reconstructed surface is closer to the originale one for m = n = 16 than for m = n = 8.

Application to medical imaging

Medical imaging provides an ideal field to test the validity of our quasi-interpolation method. For example, the left ventricle of the human heart is a muscle hollow area that can be considered closed. In this regard, we consider 1024 points of the real surface of the left ventricle. So that we can reconstruct the closed surface associated with the left ventricle using our quasi-interpolant Q (see Figure 5). We first convert the data on the rectangle H using the application χ. Then, we apply the method of least squares that distributes these data evenly over the rectangle H (see Figure 4). Finally, we reconstruct the closed surface associated with these data using the quasiinterpolants Q.

n+3 i=1 B 4 i

 4 (θ) = 1 and the family {B4

Fig. 1 .

 1 Fig. 1. The graph of the closed surface associated with f .

Fig. 2 .Fig. 3 .

 23 Fig. 2. The graph of the closed surface associated with Qf for n = 8 and m = 8.

Fig. 4 .

 4 Fig. 4. Surface mesh on the rectangle H obtained by using the least squares method.

Fig. 5 .

 5 Fig. 5. The closed surface associated with the left ventricle using our quasi-interpolant Q.