We study metric properties of convex bodies B and their polars B^o, where B is the convex hull of an orbit under the action of a compact group G. Examples include the Traveling Salesman Polytope in polyhedral combinatorics (G=S_n, the symmetric group), the set of non-negative polynomials in real algebraic geometry (G=SO(n), the special orthogonal group), and the convex hull of the Grassmannian and the unit comass ball in the theory of calibrated geometries (G=SO(n), but with a different action). We compute the radius of the largest ball contained in the symmetric Traveling Salesman Polytope, give a reasonably tight estimate for the radius of the Euclidean ball containing the unit comass ball and review (sometimes with simpler and unified proofs) recent results on the structure of the set of non-negative polynomials (the radius of the inscribed ball, volume estimates, and relations to the sums of squares). Our main tool is a new simple description of the ellipsoid of the largest volume contained in B^o.
We give a Bézout type inequality for mixed volumes, which holds true for any convex bodies. The key ingredient is the reverse Khovanskii-Teissier inequality for convex bodies, which was obtained in our previous work and inspired by its correspondence in complex geometry.
Any convex polytope whose combinatorial automorphism group has two orbits on the flags is isomorphic to one whose group of Euclidean symmetries has two orbits on the flags (equivalently, to one whose automorphism group and symmetry group coincide.) Hence, a combinatorially two-orbit convex polytope is isomorphic to one of a known finite list, all of which are 3-dimensional: the cuboctahedron, icosidodecahedron, rhombic dodecahedron, or rhombic triacontahedron. The same is true of combinatorially two-orbit normal face-to-face tilings by convex polytopes.
We classify the convex polytopes whose symmetry groups have two orbits on the flags. These exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, and their duals. The combinatorially regular two-orbit convex polytopes are certain 2n-gons for each n > 1. We also classify the face-to-face tilings of Euclidean space by convex polytopes whose symmetry groups have two flag orbits. There are finitely many families, tiling one, two, or three dimensions. The only such tilings which are also combina…
The concept of separation by hyperplanes is fundamental for convex geometry and its tropical (max-plus) analogue. However, analogous separation results in max-min convex geometry are based on semispaces. This paper answers the question which semispaces are hyperplanes and when it is possible to classically separate by hyperplanes in max-min convex geometry.
We prove the reversed Alexandrov-Fenchel inequality for mixed Monge-Ampère masses of plurisubharmonic functions, which generalizes a result of Demailly and Pham. As applications to convex geometry, this gives a complex analytic proof of the reversed Alexandrov-Fenchel inequality for mixed covolumes, which generalizes recent results in convex geometry of Kaveh-Khovanskii, Khovanskii-Timorin, Milman-Rotem and R. Schneider on reversed (or complemented) Brunn-Minkowski and Alexandrov-Fenchel inequalities. Also for toric plurisubharmonic functions in the Cegrell class, we confirm Demailly's conject…
Some conjectures and open problems in convex geometry are presented, and their physical origin, meaning, and importance, for quantum theory and generic statistical theories, are briefly discussed.
Spherical localisation is a technique whose history goes back to M.Gromov and V.Milman. It's counterpart, the Euclidean localisation is extensively studied and has been put to great use in various branches of mathematics. The purpose of this paper is to quickly introduce spherical localisation, as well as demonstrate some of its applications in convex and metric geometry.
We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of the simplexe's geometry.
We study the facial structure and Carathéodory number of the convex hull of an orbit of the group of rotations in R^3 acting on the space of pairs of anisotropic symmetric 3\times 3 tensors. This is motivated by the problem of determining the structure of some proteins in aqueous solution.
Sumset estimates, which provide bounds on the cardinality of sumsets of finite sets in a group, form an essential part of the toolkit of additive combinatorics. In recent years, probabilistic or entropic analogs of many of these inequalities were introduced. We study analogues of these sumset estimates in the context of convex geometry and Lebesgue measure on ${\mathbb R}^n$. First, we observe that, with respect to Minkowski summation, volume is supermodular to arbitrary order on the space of convex bodies. Second, we explore sharp constants in the convex geometry analogues of variants of the …
We give a short argument that for some C > 0, every n-dimensional Banach ball K admits a 256-round subquotient of dimension at least C n/(log n). This is a weak version of Milman's quotient of subspace theorem, which lacks the logarithmic factor.
We present a survey article about the geometry of convex bodies on the $d$-dimensional sphere $S^d$. We concentrate on the results based on the notion of the width of a convex body $C \subset S^d$ determined by a supporting hemisphere of $C$. Important tools are the lunes containing $C$. The supporting hemispheres take over the role of the supporting half-spaces of a convex body in Euclidean space, and lunes the role of strips. Also essential is the notion of thickness of $C$, i.e, its minimum width. In particular, we describe properties of reduced spherical convex bodies and spherical bodies …
We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colo…
The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.
In this paper we deal with a general type of integral formulas of the visual angle, among them those of Crofton, Hurwitz and Masotti, from the point of view of Integral Geometry. The purpose is twofold: to provide an interpretation of these formulas in terms of integrals of densities with respect to the canonical measure in the space of pairs of lines and to give new simpler proofs of them.
If $Ω$ is the interior of a convex polygon in $\mathbb{R}^{2}$ and $f,g$ two asymptotic geodesics, we show that the distance function $d\left(f\left(t\right),g\left(t\right)\right)$ is convex for $t$ sufficiently large. The same result is obtained in the case $\partial Ω$ is of class $C^{2}$ and the curvature of $\partial Ω$ at the point $f\left(\infty\right)=g\left(\infty\right) $ does not vanish. An example is provided for the necessity of the curvature assumption.
A theorem of Tits - Vinberg allows to build an action of a Coxeter group $Γ$ on a properly convex open set $Ω$ of the real projective space, thanks to the data $P$ of a polytope and reflection across its facets. We give sufficient conditions for such action to be of finite covolume, convex-cocompact or geometrically finite. We describe an hypothesis that make those conditions necessary. Under this hypothesis, we describe the Zariski closure of $Γ$, find the maximal $Γ$-invariant convex, when there is a unique $Γ$-invariant convex, when the convex $Ω$ is strictly convex, when we can find a $Γ$-…
The paper contains a characterization of compact groups $G\subseteq\GL(V)$, where $V$ is a finite dimensional real vector space, which have the following property \SP{}: the family of convex hulls of $G$-orbits is a semigroup with respect to the Minkowski addition. If $G$ is finite, then \SP{} holds if and only if $G$ is a Coxeter group; if $G$ is connected then \SP{} is true if and only if $G$ is polar. In general, $G$ satisfies \SP{} if and only if it is polar and its Weyl group is a Coxeter group.
We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of $c_0$ and superreflexivity are discussed.
The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three and that in higher dimension it is a lower bound of the entropy. As a corollary we solve the entropy upper bound conjecture in dimension three and give a new proof in dimension two from the one found in Berck-Bernig-Vernicos (arXiv:0810.1123v2, published).
A convex quadrangular pyramid $ABCDE$, where $ABCD$ is the base and $E$ -- the apex, is called \emph{strongly flexible}, if it belongs to a continuous family of pairwise non-congruent quadrangular pyramids that have the same lengths of corresponding edges. $ABCDE$ is called \emph{strongly rigid}, if such family does not exist. We prove the strong rigidity of convex quadrangular pyramids and prove that strong rigidity fails in the self-intersecting case. Let $L=\{l_1,\ldots,l_8\}$ be a set of positive numbers, then a \emph{realization} of $L$ is a convex quadrangular pyramid $ABCDE$ such, that …
In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative formulas extend to arbitrary continuous and translation invariant valuations. Then, we discuss applications to Poisson particle processes and Boolean models which contain, as a special case, some new results for flag measures.
In this paper we show the following result: if C is an n-dimensional 0-symmetric convex compact set, $f:C\rightarrow[0,1)$ is concave, and $g:[0,1)\rightarrow[0,1)$ is not identically zero, convex, with g(0)=0, then \[ \frac{1}{|C|}\int_C g(f(x))dx \leq \frac12 \int_{-1}^1g(f(0)(1+t))dt, \] where |C| denotes the volume of C. If g? is strictly convex, equality holds if and only if f is affine, C is a generalized symmetric cylinder and f becomes 0 at one of the basis of C. We exploit this inequality to answer a question of Francisco Santos on estimating the volume of a convex set by means of the…
This paper presents connections between Gromov's work on isoperimetry of waists and Milman's work on the $M$-ellipsoid of a convex body. It is proven that any convex body $K \subseteq \mathbb{R}^n$ has a linear image $\tilde{K} \subseteq \mathbb{R}^n$ of volume one satisfying the following waist inequality: Any continuous map $f:\tilde{K} \rightarrow \mathbb{R}^{\ell}$ has a fiber $f^{-1}(t)$ whose $(n-\ell)$-dimensional volume is at least $c^{n-\ell}$, where $c > 0$ is a universal constant. In the specific case where $K = [0,1]^n$ it is shown that one may take $\tilde{K} = K$ and $c = 1$, con…
Let $K$ be an $n$-dimensional symmetric convex body with $n \ge 4$ and let $K\dual$ be its polar body. We present an elementary proof of the fact that $$(\Vol K)(\Vol K\dual)\ge \frac{b_n^2}{(\log_2 n)^n},$$ where $b_n$ is the volume of the Euclidean ball of radius 1. The inequality is asymptotically weaker than the estimate of Bourgain and Milman, which replaces the $\log_2 n$ by a constant. However, there is no known elementary proof of the Bourgain-Milman theorem.
The intrinsic volumes of a convex body are fundamental invariants that capture information about the average volume of the projection of the convex body onto a random subspace of fixed dimension. The intrinsic volumes also play a central role in integral geometry formulas that describe how moving convex bodies interact. Recent work has demonstrated that the sequence of intrinsic volumes concentrates sharply around its centroid, which is called the central intrinsic volume. The purpose of this paper is to derive finer concentration inequalities for the intrinsic volumes and related sequences. T…
In this work we study a class of random convex sets that "interpolate" between polytopes and zonotopes. These sets arise from considering a $q^{th}$-moment ($q\geq 1$) of an average of order statistics of $1$-dimensional marginals of a sequence of $N\geq n$ independent random vectors in $\mathbb R^n$. We consider the random model of isotropic log-concave distributions as well as the uniform distribution on an $\ell_p^n$-sphere ($1\leq p < \infty$) with respect to the cone probability measure, and study the geometry of these sets in terms of the support function and mean width. We provide asymp…
