The Santal\'o point for the Holmes-Thompson boundary area

We explore the notion of Santal\'o point for the Holmes-Thompson boundary area of a convex body in a normed space. In the case where the norm is $C^1$, and in the case where unit ball and convex body coincide, we prove existence and uniqueness. When the normed space has a smooth positively curved unit ball, we exhibit a dual Santal\'o point expressed as an average of centroids of projections of the dual body.


introduction
Several decades ago Santaló studied in [Santaló 1949] the functional (1.1) associated to any convex body K of R n . Here | · | denotes the Lebesgue measure of R n , and A • = {y ∈ R n | x, y = n i=1 x i y i ≤ 1 ∀x ∈ A} is the polar set of a convex body A. He found that this functional is proper and strictly convex. Consequently, there exists a unique minimizing point s(K) in the interior of K, known nowadays as the Santaló point of K. Along the way, Santaló computed the derivative of (1.1) and showed that where c(K • ) = K • x dx is the centroid of the polar body K • and v any vector in R n . Therefore s(K) lies at the origin if and only if c(K • ) lies at the origin. Determining the Santaló point of a given convex body can be a difficult question, so this characterization is particularly useful.
As the Lebesgue measure coincides with the n-dimensional Hausdorff measure of R n , it is natural to consider the following functional similar to (1.1) where H n−1 denotes the (n − 1)-dimensional Hausdorff measure, and look for possible minimizing points.
More generally, let V be an n-dimensional real vector space. To any convex body B in V that contains the origin in its interior is associated a unique (asymmetric) norm · B whose unit ball is precisely B. The restriction of · B to any hypersurface defines a Finsler metric whose corresponding Holmes-Thompson (n − 1)-volume (or area in short) will be denoted by A B (·). If K is a convex body in (V, · B ), we can consider its dual body K * = {p ∈ V * | p(x) ≤ 1 ∀x ∈ K} as a convex body of the dual normed vector space (V * , · B * ) and focus on the associated Holmes-Thompson area of its boundary sphere, that is the quantity A B * (∂K * ). In the special case where V = R n and B is the Euclidean unit ball, we have A B * (∂K * ) = H n−1 (∂K • ). Therefore functional (1.3) turns out to be a particular case of the functional A minimizing point of this functional will be called a Santaló point of K for the Holmes-Thompson area in (V, · B ).
In this paper we first prove existence and uniqueness of Santaló points for the Holmes-Thompson area in any normed vector space whose unit ball is of class C 1 .
Theorem 1.1. Let K be a convex body in a finite-dimensional real normed vector space (V, · B ) whose unit ball B is of class C 1 . The functional is strictly convex and proper on the interior of K.
In particular there exists a unique minimizing point S B (K) ∈ int(K) for this functional.
In case the unit ball is no longer of class C 1 , the above functional is still convex (and proper) on the interior of K, but strict convexity is no longer guaranteed. Therefore, Santaló points do exist but are no longer necessarily unique. In section 3, our study goes far beyond the C 1 -regularity, and we give precise necessary and sufficient conditions for the uniqueness. For example, when the unit ball B is a polytope there is always a convex body K with an infinite set of Santaló points, as illustrated in the concrete example 3.5. On the bright side, uniqueness is always ensured without any regularity assumption in the special case where unit ball and convex body coincide. Using the duality formula A B * (∂K * ) = A K (∂B) discovered by [Holmes & Thompson 1979], we restate this result as follows.
Theorem 1.2. Let (V, · B ) be any finite-dimensional real normed vector space. The functional admits a unique minimizing point S B (B).
Rephrasing the theorem above, there exists a unique translation of the unit ball of a given normed vector space minimizing its own Holmes-Thompson boundary area.
In addition, by invariance properties of the Holmes-Thompson area, the map B → S B (B) is affinely equivariant, that is S T B (T B) = T (S B (B)) for any invertible affine map T . Therefore this map defines a new affine-invariant point in the sense of [Grünbaum 1963]. See subsection 3.3 for more details.
In order to characterize S B (K), we study the first variation of the above functional. Assuming that the normed vector space (V, · B ) is Minkowski, i.e. ∂B is smooth and positively curved (that is, with strictly positive sectional curvature), we obtain an interesting formula closely ressembling (1.2) and involving some classical notions from affine differential geometry. Given x ∈ ∂B, recall that the affine normal line is a canonical 1-dimensional vector subspace N x ⊂ V transverse to T x = T x ∂B, which behaves equivariantly under linear transformations of ∂B. We thus have the decompositions x given by φ → φ| Tx , and the projection π x : V * → N ⊥ x with kernel ker π x = T ⊥ x . Pick an arbitrary smooth measure µ on ∂B and consider the Lebesgue measure µ x on T x , and its dual measure (µ x ) * on T * x . Let ν x be the Lebesgue measure on N ⊥ Definition 1.3. Let K be a convex body in a Minkowski space (V, · B ). Using the previous notation, we define Note that the inner integral is the centroid of the projection π x (K * ) with respect to ν x and belongs to V * . It is easy to see that C B (K * ) does not depend on the choice of µ.
Our third main result is the following.
Theorem 1.4. Let K be a convex body in a Minkowski space (V, · B ). Then for any In particular S B (K) lies at the origin of V if and only if C B (K * ) lies at the origin of V * .
Here ε n−1 denotes the Euclidean volume of an (n − 1)-dimensional unit Euclidean ball and ·, · the canonical duality pairing. Note the similarity with formula (1.2).
In the particular case of a convex body in an Euclidean space, both theorems sum up into the following.
Corollary 1.5. Given a convex body K in R n , there exists a unique point S(K) ∈ int(K) minimizing the functional Furthermore, this point lies at the origin if and only if the following average of centroids of orthogonal projections of its polar body lies at the origin.
Here π x ⊥ denotes the orthogonal projection onto the hyperplane orthogonal to x, and σ, λ denote respectively the canonical measure on S n−1 and the Lebesgue measure on x ⊥ induced by the standard Euclidean structure.
This paper is organized as follows. In section 2 we collect some material on the Holmes-Thompson volume, and present a Crofton formula for the Holmes-Thompson area of a hypersurface in a Minkowski space due to [Álvarez Paiva 1998]. This formula will be decisive to show (1.4). In section 3 we prove Theorems 1.1 and 1.2. The next three sections are devoted to the proof of Theorem 1.4. The strategy consists of using the above mentioned Crofton formula which describes the Holmes-Thompson area of ∂K * as an integral on the set of lines intersecting K * . In order to compute its first variation under translations, we first use the boundary sphere ∂B to construct a parametrization space for the oriented affine lines of the dual vector space in terms of intersections of affine hyperplanes. We then show how to rewrite the measure involved inÁlvarez Paiva's formula in this new space. The effect of translations is then simple to describe which allows us to compute the first variation of the area. So after shortly recalling in section 4 some classical notions of equiaffine differential geometry, we define the parametrization space in section 5 and show how to rewrite the measure of lines in this new setup. In the last section 6 we compute the first variation of the Holmes-Thompson area and obtain formula (1.4).
Acknowledgements. We would like to thank the referee for valuable comments.

Background on the Holmes-Thompson notions of volume and area
In this section we collect some material on the Holmes-Thompson notions of volume and area. The two major tools we will use in the sequel are a Crofton type formula due toÁlvarez Paiva and presented in Proposition 2.1, and a duality formula due to Holmes & Thompson stated in Proposition 2.3.
2.1. Holmes-Thompson volume on continuous Finsler manifolds. Let M be a smooth manifold M of dimension k endowed with a continuous Finsler metric F : T M → [0, ∞), that is a continuous function which is positively homogenous of degree 1 outside the zero section and such that for all x ∈ M the subsets of are convex bodies containing the origin in their interior (or equivalently, F is a continuous function whose restriction to each tangent space T x M is a norm, possibly asymmetric). For any measurable set A ⊂ M denote by D * (A, F ) the unit co-disc bundle over A consisting of the disjoint union of the dual bodies D x (M, F ) * ⊂ (T x M ) * for x ∈ A. The Holmes-Thompson volume of the measurable set A in (M, F ) is then defined as the following integral: where ω M denotes the standard symplectic form on the cotangent bundle T * M . If ρ denotes any non-vanishing k-density on M , then where | · | * ρx is the dual Lebesgue measure on T * x M of the Lebesgue measure | · | ρx on T x M associated to ρ. More precisely, | · | ρx is the Lebesgue measure on T x M normalized to give volume 1 on a parallelotope spanned by a base v 1 , . . . , v k of T x M for which ρ x (v 1 , . . . , v k ) = 1, and | · | * ρx is the Lebesgue measure on T * x M normalized to give volume 1 for the parallelotope spanned by the covectors ξ 1 , . . . , ξ k ∈ T * x M forming a base which is dual to (v 1 , . . . , v k ).

2.2.
Holmes-Thompson volume and area in normed spaces. Let B be a convex body in an n-dimensional real vector space V that contains the origin as an interior point. The associated norm · B defines a continuous Finsler metric on the manifold V whose Holmes-Thompson volume satisfies for any compact set A the equality where | · | and | · | * are respectively the measure and its dual measure associated to some translationinvariant density on V . We will denote shortly V(A, · B ) by V B (A).
In the case where V = R n we find that V B (A) = |A| |B • |/ε n . In particular the Finsler volume V B (B) of the convex body B itself coincides with its volume product up to the constant 1/ε n .
It turns out that the associated norm · B also permits to define for any hypersurface M ⊂ V (and, more generally, for any smooth submanifold of V ) a continuous Finsler metric F B : T M → [0, ∞) by simply setting F B (x, v) = v B . We will denote shortly V(M, F B ) by A B (M ) and speak of the Holmes-Thompson area of M . This notion of area extends to boundaries of convex bodies. For this recall that the set of singular boundary points (those for which the supporting hyperplane is not uniquely defined) of a convex body K in V is of (n − 1)-dimensional Hausdorff measure zero, see [Reidemeister 1921]. Therefore the formula where ρ denotes any translation-invariant (n − 1)-density on V makes sense when M = ∂K is the boundary of a convex body K. Moreover, if K n → K is a sequence of smooth convex bodies that converges in the Hausdorff metric to a convex body K, then A B (∂K n ) → A B (∂K) (see Remark 2.2).
2.3. Crofton formula for hypersurfaces in Minkowski spaces. In case the normed space (V, · B ) has smoothness and strict convexity properties, the Holmes-Thompson area of hypersurfaces admits a Crofton formula due to Alvarez-Paiva (cf. [Álvarez Paiva 1998]) that we present next. We refer to [Schneider 2006] for a very complete survey on the integral geometry of normed spaces.
Let B be a convex body of V that contains the origin as an interior point, and suppose ∂B is smooth and positively curved. Then the associated normed space (V, · B ) is said to be Minkowski. In that case its dual body also has smooth and positively curved boundary, and the dual Legendre transform L * : ∂B * → ∂B is a well defined diffeomorphism. Recall that L * (p) is defined as the unique x ∈ ∂B such that p(x) = 1. Denote by G + (V ) the space of oriented affine lines in V whose elements are denoted by x + v + where x is a point in V and v + is the 1-dimensional vector subspace of V spanned and oriented by a vector v. In particular according to this notation x + v + = y + w + if and only if there exists (t, s) ∈ R × R >0 such that y = x + tv and w = sv. Now consider the projection map Recall that ω V denotes the standard symplectic form of T * V and denote by i : V × ∂B * → T * V the canonical inclusion. It is well known that there exists a unique symplectic form ω B on G + (V ) such that π * ω B = i * ω V , see [Arnold & Givental 1990, Besse 1978. Here is the Crofton formula associated to this symplectic form.
Proposition 2.1 (Álvarez Paiva). The Holmes-Thompson area of a compact immersed hypersurface M in a Minkowski space (V, · B ) satisfies the following formula: Applying this formula to the boundary of a smooth convex K in V we obtain the following formula: By continuity we see that this formula still holds when the convex body K is not necessarily smooth.
Proposition 2.1 was first stated in [Álvarez Paiva 1998] (see [Álvarez & Fernandes 1998, Theorem 3.1]) more generally for immersed hypersurfaces in reversible Finsler manifolds whose space of oriented geodesics is a smooth manifold. We include the proof here for the reader's convenience as well to check that it is still true in our context without the reversibility assumption.
Therefore, by the coarea formula Here |η| denotes the density given by the absolute value of a top differential form η. Now observe that P * M ω M = i * ω V (as both are the exterior derivative of the tautological one-form) which implies, using the identity i * ω V = π * ω B and the coarea formula again, that Observe that we obtain by the way the following formula for Holmes-Thompson area: Remark 2.2. We easily check from the above formula that if K n , B n are sequences of convex bodies that converge in the Hausdorff metric to K, B respectively, then A Bn (∂K n ) → A B (∂K).

Holmes-Thompson duality formula.
To conclude this section let us recall the following duality principle for the Holmes-Thompson area of convex bodies.

Proposition 2.3 (Holmes & Thompson)
. Let K and B be two convex bodies in V that contain the origin in their interior. Then Proof. This was first observed by Holmes and Thompson in [Holmes & Thompson 1979]. Here is a quick argument: where we have abusively denoted i * ω V by ω V .

Strict convexity and properness
In this section we prove both Theorem 1.1 and Theorem 1.2 which we merge in a same statement as follows using Proposition 2.3.
is strictly convex and proper on the interior of K.
In particular there exists a unique minimizing point S B (K) ∈ int(K) for the Holmes-Thompson area of the boundary sphere of (K − x) * in (V * , · B * ).
We split the proof into two subsections. The first concerns the strict convexity of the functional above. The second proves the properness of the functional, using an isoperimetric inequality between Holmes-Thompson notions of volume and area. In a third subsection we explain why the map B → S B (B) gives rise to an affine-invariant point. Finally, in the last subsection we further explore conditions which ensure the strict convexity beyond conditions (i) and (ii).
3.1. Proof of the strict convexity. After fixing some isomorphism V R n , we can use the standard Euclidean structure to identify the dual body A * with the polar body A • of any convex body A containing the origin as an interior point, and rewrite using formula (2.1). Here we have denoted by • dλ n−1 the (n − 1)-dimensional volume element induced on hypersufaces by the Euclidean structure • |C| n−1 = C dλ n−1 the (n − 1)-Hausdorff measure of any compact domain C contained in some hyperplane.
Recall that a point y in the boundary of a convex body B is said to be regular if there is a unique supporting hyperplane y + H(y, B) of B at y. The set reg B ⊂ ∂B of regular points has (n − 1)-Hausdorff measure H n−1 (reg B) = H n−1 (∂B) (see [Reidemeister 1921]), and the map is continuous (cf. [Schneider 2013, Lemma 2.2.12]). Here Gr k (R n ) denotes the set of all vector k-planes in R n . Hence, the push-forward of the (n − 1)-Hausdorff measure H n−1 by this map is well defined and we denote by µ ∂B this measure on Gr n−1 (R n ). The measure µ ∂B coincides with the push-forward of the classical surface area measure of B (which is a measure on the sphere S n−1 ) using the standard two folding map S n−1 → Gr n−1 (R n ).
Consider the functional which is continuous as a composition of continuous maps.
Proof. By [Schneider 2013, eqs. (1.52) & (1.53)], in an m-dimensional Euclidean vector space (W, ·, · ), the Lebesgue measure of any convex body C can be computed using the formula where S m−1 (W ) denotes the unit sphere in (W, ·, · ) and h C • is the support function of the polar body C • , defined for any is concave as for any λ ∈ (0, 1) and The function t → 1/t n−1 being strictly convex on (0, ∞), we get that the function is also convex as an integral of convex functions.
For any x 1 = x 2 ∈ int(K) let us define Strcvx K (x 1 , x 2 ) ⊂ Gr n−1 (R n ) as the set of hyperplanes H such that is an open set by continuity of f K . Furthermore, for a fixed H, we know by [Santaló 1949] that for any Here ∆ denotes the diagonal subset. We denote by Cylpair(K) ⊂ int(K)×int(K)\∆ the subset of cylindrical pairs associated to K. The term cylindrical pair is justified by the following result.
Lemma 3.3. For any (x 1 , x 2 ) ∈ Cylpair(K), we can find a hyperplane H ∈ Gr n−1 (R n ) and a convex body C of H such that for λ = 1/2 and thus for all λ ∈ (0, 1). It follows by the proof of Lemma 3.2 that ∀u ∈ S n−2 (H) and ∀λ ∈ [0, 1] The following result describes how cylindrical pairs of K should interact with the support of µ ∂B to ensure strict convexity for the Holmes-Thompson area. Recall that the support supp(µ) of a Borel measure µ is the intersection of all closed sets of total measure.
x 2 ) = Gr n−1 (R n ) in this case. So we can replace "for every pair (x 1 , x 2 ) ∈ Cylpair(K)" by "for every x 1 = x 2 ∈ int(K)" in the proposition above.
Proof. The integral of a non-negative continuous function against a Borel measure is positive if and only if the support of the measure meets the interior of the support of the function. Therefore, fixing is positive if and only if Strcvx K (x 1 , x 2 ) meets supp(µ ∂B ).
We are now ready to prove the strict convexity part of Theorem 3.1.
Proof of the strict convexity in Theorem 3.1. (i) Suppose first that ∂B is of class C 1 . Thus supp(µ ∂B ) = Gr n−1 (R n ) which implies that the map x → A K−x (∂B) is always strictly convex on int(K) according to Proposition 3.4.
By Lemma 3.3, given any (x 1 , x 2 ) ∈ Cylpair(B), there exists H ∈ Gr n−1 (R n ) and C a convex body in H such that (H + [x 1 , . Let µ ∂C be the push-forward measure of the surface area measure of C by the quotient map S n−2 (H) → Gr n−2 (H). Take any G ∈ supp(µ ∂C ), and consider the hyperplane H = G ⊕ span(x 2 − x 1 ). Since x 2 − x 1 ∈ H , we know by [Santaló,1949] that H ∈ Strcvx B (x 1 , x 2 ). Let N be any open neighborhood of H in Gr n−1 (R n ). Let us prove that µ ∂B (N ) > 0. Consider the map which is continuous. Since ι −1 (N ) is open and contains G ∈ supp(µ ∂C ) we have µ ∂C (ι −1 (N )) > 0. Let U be the set of regular points of ∂B with supporting hyperplane inside N , and let V ⊂ ∂C consist of the regular points of ∂C with supporting hyperplane in ι −1 (N ). By [Rockafellar 1997, Theorems 25.1 and 25.5], there exists an injective C 1 -map ϕ defined on some measurable set D ⊂ R n−2 and such that ϕ(D) = V . Setφ Observe that the Jacobians of ϕ andφ satisfy the inequality Jac xφ ≥ Jac x ϕ · sin α where α denotes the angle between x 2 − x 1 and H. Therefore we have as claimed. This implies that H ∈ supp(µ ∂B ). Since also H ∈ Strcvx B (x 1 , x 2 ), this proves strict convexity in case (ii).
Remark 3.5. We now present an example of two convex bodies K and B in the plane R 2 such that the functional x → A K−x (∂B) is convex but not strictly convex: Observe that supp(µ ∂B ) consists of the two directions x = y, x = −y each of weight 2 √ 2. It is easy to check that the interior of the shaded area corresponds to points (x, y) where the functional t → A K−(x+t,y) (∂B) is constant for small values of t. By symmetry we deduce that this functional achieves its minimum on any point lying on the black segment.
3.2. Proof of the properness. In order to prove Theorem 3.1, it remains to show that the map is proper on the interior of K. This holds without any assumption on the convex body B, and easily follows from the well-known fact that the map x → V K−x (B) is proper on the interior of K and the following generalization of the classical isoperimetric inequality to asymmetric Minkowski spaces.
Proposition 3.6 (Isoperimetric inequality for Holmes-Thompson volume). Let B and K be two convex bodies of an n-dimensional real vector space V . Suppose that K contains the origin as an interior point. Then This inequality is stated in [Thompson 1996, Theorem 6.6.4] in the case where K is symmetric. The proof straightforwardly generalizes to the non-symmetric case so we briefly survey the arguments.
Proof. After fixing some isomorphism V R n , we identify the dual body A * of any convex body A containing the origin as an interior point with its polar body A • . First recall that (compare with [Thompson 1996, Theorem 5.2

.2])
where V denotes the mixed volume and I K the isoperimetrix convex body defined as the unique symmetric convex body with support function (3.3) Observe that I K is a zonoid according to [Schneider 2013, Formula 5.80 and Theorem 3.5.3].
Next we check that Indeed as A K (∂I K ) = n|I K | and V K (B) = |B||K • |/ε n , the above inequality is equivalent to V (B[n − 1], I K ) n ≥ |I K ||B| n−1 which directly follows from Minkowski's inequality, see [Schneider 2013, Theorem 7.2.1].
Now recall that the normalized isoperimetrix convex bodyĨ K is defined as the unique dilated of the isoperimetrix satisfying n V K (Ĩ K ) = A K (∂Ĩ K ). Therefore Besides it is easy to see that in factĨ K = ε n · I K /|K • |. Using equation (3.3), we also see that ε n−1 · h I K is precisely the support function of the projection body ΠK • . Consequently we find the identityĨ K = (ε n /ε n−1 |K • |) · ΠK • . Therefore which implies together with Petty's projection inequality [Petty 1971], see [Schneider 2013, Formula 10.86], that 4n) n n!ε n using Reisner optimal lower bound on the volume product for zonoids [Reisner 1985].
The properness statement in Theorem 3.1 is now a consequence of the following simple fact.
Proposition 3.7. Let B k → B be a convergent sequence of convex bodies in R n , and assume that 0 ∈ int(B k ) for each k and 0 ∈ ∂B. Then |B • k | → ∞.
Proof. Given a convex body K, let us say that a parallelotope circumscribes K if K ⊂ P and every face of P is contained in a supporting hyperplane of K. Without loss of generality assume that B is supported by a coordinate hyperplane at 0. Let P k (resp. P ) be the parallelotope circumscribing B k (resp. B), and with all faces parallel to the coordinate hyperplanes. Then P • n is a sequence of polytopes with vertices contained in the coordinate lines. One of these vertices tends to infinity while the others converge to points different from the origin. It follows that

3.3.
A new affine-invariant point. Let us start by recalling the definition of an affine-invariant point (see [Grünbaum 1963]): this is a map f : K(R n ) → R n where K(R n ) denotes the space of convex bodies in R n satisfying the following two conditions • for every inversible affine map Φ : R n → R n and every convex body B ∈ K(R n ), one has f (Φ(B)) = Φ(f (B)). • f is continuous with respect to the Hausdorff metric.
We will prove the following. Proof. Using (2.2) we easily check that given any pair K, B ∈ K(R n ) we have A LK (∂LB) = A K (∂B) for every invertible linear map L : R n → R n . So if T : R n → R n is an invertible affine map, we get that for all x ∈ int(B). Our Santaló point thus satisfies the equality from which we deduce that the map f S fulfills the first condition.
In order to check the second condition, we argue as follows. First define a function and F S (B, x) = +∞ otherwise. Observe that this function is continuous. Indeed, since F S (B, x) = F S (B − x, 0), we only need to check continuity with respect to B. As sequentially continuous maps defined on metric spaces are continuous, this follows from Remark 2.2 and Propositions 3.6 and 3.7.
Given a convex body B, fix K 0 ∈ K(R n ) such that B ⊂ int(K 0 ). By the Blaschke selection theorem, the subset V : 3.4. Further study of the strict convexity. Next, we describe an optimal property on B that ensures strict convexity for any convex body K. For this, let us define for any L ⊂ Gr 1 (R n ) the subset Star(L) := {H ∈ Gr n−1 (R n ) | L ⊂ H}.
Proposition 3.9. Let B be a convex body. The map x ∈ int(K) → A K−x (∂B) is strictly convex for any convex body K if and only if supp(µ ∂B ) ∩ Star(L) = ∅ for any L ∈ Gr 1 (R n ).
Proof. Recall that as already observed, for a fixed H and K, we know by [Santaló 1949] that for any In particular, for any x 1 = x 2 ∈ int(K), we have that Star(span(x 2 −x 1 )) ⊂ Strcvx K (x 1 , x 2 ). So if supp(µ ∂B )∩Star(L) = ∅ for any L ∈ Gr 1 (R n ), we directly get that supp(µ ∂B ) ∩ Strcvx K (x 1 , x 2 ) = ∅ for any convex body K and x 1 = x 2 ∈ int(K). By Proposition 3.4, it ensures that the map x ∈ int(K) → A K−x (∂B) is always strictly convex. Now suppose that supp(µ ∂B ) ∩ Star(L) = ∅ for some L ∈ Gr 1 (R n ). Proof of the Lemma. Fix an open neighborhood U of Star(L) in Gr n−1 (R n ). We construct a convex body K such that for some For this consider the (possibly empty) compact set U c and let K be any infinite cylinder with direction L, with a compact convex base and that contains 0 in its interior. Now fix ε 0 > 0 and a unit vector v ∈ L. As U c is compact and disjoint from Star(L), we can consider a finite subcylinder K of K delimited by two affine hyperplanes whose underlying directions are not in Star(L) and such that (H + εv) ∩ K = (H + εv) ∩ K for any H ∈ U c and any |ε| ≤ ε 0 . By taking x 1 = −ε 0 v and Lemma 3.10 applied to the open neighborhood U := supp(µ ∂B ) c implies that there exist K and x 1 = x 2 ∈ int(K) such that Strcvx K (x 1 , x 2 ) ⊂ supp(µ ∂K ) c and by Proposition 3.4 we are done.
To conclude this section, we describe an optimal property on K that ensures strict convexity for any convex body B.
Proposition 3.11. Then the map x → A K−x (∂B) is strictly convex on int(K) for any convex body B if and only if K has no cylindrical directions, that is Cylpair(K) = ∅.
Proof. The fact that the condition Cylpair(K) = ∅ is sufficient follows directly from 3.4. To prove that it is also necessary, let K be a convex body with Strcvx K (x 1 , x 2 ) = Gr n−1 (R n ) for some cylindrical pair (x 1 , x 2 ). We will construct B such that supp(µ ∂B ) ∩ Strcvx K (x 1 , x 2 ) = ∅. For this we need the following lemma.
Lemma 3.12. If Strcvx K (x 1 , x 2 ) c is nonempty, then it has a nonempty interior.
Proof. According to Lemma 3.3, if Strcvx K (x 1 , x 2 ) = Gr n−1 (R n ), we can find a hyperplane H 0 ∈ Gr n−1 (R n ) and a convex body From this lemma, it follows that if Strcvx K (x 1 , x 2 ) = Gr n−1 (R n ), then Strcvx K (x 1 , x 2 ) c contains n hyperplanes defined by linearly independent covectors. So, for any parallelotope B whose face directions are precisely these n hyperplanes, we have that supp(µ ∂B ) ∩ Strcvx K (x 1 , x 2 ) = ∅, and so we can conclude by Proposition 3.4.

Equiaffine differential geometry
We now introduce classical notions from equiaffine geometry and some notations that will be used to prove Theorem 1.4 in the next two sections.
From here on, let us suppose the n-dimensional vector space V endowed with a non-trivial alternate multilinear n-form, which we denote by det. Let M ⊂ V be a hypersurface and take a vector field Ξ transverse to M . Then, for every pair of tangent vector fields X, Y on M , we can decompose the flat connection ∇ of V as ∇ X Y = ∇ M X Y + g(X, Y ) · Ξ (4.1) where ∇ M is an affine torsion-free connection on M and g is a field of symmetric bilinear forms. The hypersurface M is said to be non-degenerate if g is nowhere degenerate, a condition that does not depend on Ξ.
The pseudo-Riemannian metric g and its volume form α are called equiaffine metric and equiaffine area measure respectively. We assume from here on that g is Riemannian.

Measure of lines in terms of hyperplanes
In this section we use the boundary sphere ∂B to construct a parametrization space for the oriented affine lines of the dual vector space in terms of intersections of affine hyperplanes. We then show how to rewrite the measure involved inÁlvarez Paiva formula in this new space.
Recall that we have fixed some non-trivial alternate multilinear n-form det on the n-dimensional vector space V .
Let B be a convex body of V that contains the origin as an interior point, and suppose ∂B is smooth and positively curved. In particular its dual body is smooth, and the Legendre transform L : ∂B → ∂B * , uniquely defined by is a well defined diffeomorphism. Moreover, since ∂B has non-degenerate second fundamental form with respect to any Euclidean structure, Theorem 4.1 applies to it. We orient ∂B with the inward vector (i.e. as the boundary of V \ B) so that the equiaffine metric g is Riemannian. Associated to the hypersurface ∂B we also have well defined functions L i,j and L ∈ C ∞ (E(∂B)) (see Definition 4.2).
It is easy to check that the previous diagram commutes.
Recall that G + (V * ) is endowed with a symplectic form ω B * (see section 2.3). It will be convenient to consider on G + (V * ) the following associated volume element η B * = (−1) n(n−1) 2 ω n−1 B * and take the corresponding orientation. Our goal in this section is to compute the pull-back form G * η B * . For this, we introduce the following 2-forms.
The rest of this section is devoted to the proof of the proposition above.
5.1. Technical lemmas. Let us begin with the following computation.
Proof. Note first that Thus, modulo terms of the form The statement follows.
Secondly we prove the following identity.

Proof of Proposition Let us denote
The proposition will follow easily after proving that (1) F * Ω = F * G * η B * ; (2) i X Ω = 0 for all X ∈ ker dG.

First variation of the dual area under translations
Recall that we have chosen some non-trivial alternate multilinear n-form denoted by det on the ndimensional vector space V . Recall also that B ⊂ V is a convex body containing the origin in its interior and has smooth and positively curved boundary (i.e. the normed vector space (V, · B ) is Minkowski). Let K denote another convex body of V .
In this section we prove Theorem 1.4 by explicitly computing for any v ∈ V .
6.1. Preliminaries. Let us begin with the following simple fact.
Proof. Recall that for some q ∈ V * . Hence G(ξ) ⊂ H y if and only if y ∈ ker L(x) = span(ξ 1 , . . . , ξ n−1 ) and q, y = 1. This is equivalent to y = i λ i ξ i with i λ i = q, y = 1, which means that y is in the affine hull of the ξ i . Given Here x v ∈ ∂B is uniquely determined by the conditions that ξ i +v ∈ T xv ∂B for all i and that the basis {x v , ξ 1 +v, . . . , ξ n−1 +v} is negatively oriented. Note that −v / ∈ aff(ξ) ensures that ξ 1 +v, . . . , ξ n−1 +v are linearly independent.
To prove that this is independent of ξ, assume G(ξ) = G(ξ ) for some ξ = (x , ξ 1 , . . . , ξ n−1 ). By Lemma 6.1, each ξ i is in the affine hull of ξ 1 , . . . , ξ n−1 . Since ξ i + v is clearly also in the affine hull of the ξ j + v, we get by Lemma 6.1 that G(T v (ξ)) = G(T v (ξ )) up to orientation. Since v → x v is continuous, the orientations of G(T v (ξ)) and G(T v (ξ )) must agree or disagree for all v; and they coincide for v = 0. This shows that T v is well-defined.
Lemma 6.2. For v ∈ V and L ∈ G + (V * ) we have (6.1) Proof. Given ζ = (x, ζ 1 , . . . , ζ n−1 ) ∈ E(∂B) and a convex body Q in V , note that G(ζ) ∩ Q * = ∅ if and only if every hyperplane containing G(ζ) intersects Q * . By Lemma 6.1 this is equivalent to the fact that H y ∩ Q * = ∅ for every y in the affine hull of ζ 1 , . . . , ζ n−1 . In turn, this is equivalent to y / ∈ Q for all such y.
Let now L = G(ξ). Taking ζ = ξ and Q = K − v we get that the left hand side of (6.1) holds if and only if y / ∈ K − v for every y ∈ aff(ξ). Since 0 ∈ int(K), this is equivalent to the fact that −v / ∈ aff(ξ) and z / ∈ K for all z ∈ aff(T v (ξ)). Taking ζ = T v (ξ) and Q = K this condition is equivalent to the right hand side of (6.1).

6.2.
Proof of Theorem 1.4. Take a local section ζ : U → E(∂B) mapping each x ∈ U to a positive orthonormal basis {e 1 (x), . . . , e n−1 (x)} of T x ∂B with respect to the Riemannian metric g.
Therefore, using Proposition 6.3, we get the following formula: Let us define the map which is a diffeomorphism onto its image that satisfies G • F (λ, x) = p x (λ) + L(x) + .
In particular, we have where π x : V * → ker Ξ(x) is the linear projection with ker π x = L(x) .
Covering ∂B with local sections and using (6.2), we deduce the stated formula: q dν x (q) dµ(x), v .