We prove that the reciprocal of the volume of the polar bodies, about the Santaló point, of a {\em shadow system} of convex bodies $K_t$, is a convex function of $t$. Thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of $t$ is also investigated and is characterized under certain conditions.
We apply these results to prove exact reverse Santaló inequality for polytopes in $\rd{d}$ that have at most $d+3$ vertices.
It is proved that if $C$ is a convex body in ${\Bbb R}^n$ then $C$ has an affine image $\widetilde C$ (of non-zero volume) so that if $P$ is any 1-codimensional orthogonal projection, $$|P\widetilde C| \ge |\widetilde C|^{n-1\over n}.$$ It is also shown that there is a pathological body, $K$, all of whose orthogonal projections have volume about $\sqrt{n}$ times as large as $|K|^{n-1\over n}$.
We construct examples of two convex bodies $K,L$ in $\mathbb{R}^n$, such that every projection of $K$ onto a $(n-1)$-dimensional subspace can be rotated to be contained in the corresponding projection of $L$, but $K$ itself cannot be rotated to be contained in $L$. We also find necessary conditions on $K$ and $L$ to ensure that $K$ can be rotated to be contained in $L$ if all the $(n-1)$-dimensional projections have this property.
We obtain optimal inequalities for the volume of the polar of random sets, generated for instance by the convex hull of independent random vectors in Euclidean space. Extremizers are given by random vectors uniformly distributed in Euclidean balls. This provides a random extension of the Blaschke-Santalo inequality which, in turn, can be derived by the law of large numbers. The method involves generalized shadow systems, their connection to Busemann type inequalities, and how they interact with functional rearrangement inequalities.
We investigate a duality relation between floating and illumination bodies. The definitions of these two bodies suggest that the polar of the floating body should be similar to the illumination body of the polar. We consider this question for the class of centrally symmetric convex bodies. We provide precise estimates for $(B_p^n)$ and for centrally symmetric convex bodies with everywhere positive Gauss curvature. Our estimates show that equality of the polar of the floating body and the illumination body of the polar can only be achieved in the case of ellipsoids.
For each i = 1, ..., n constructions are given for convex bodies K and L in n-dimensional Euclidean space such that each rank i orthogonal projection of K can be translated inside the corresponding projection of L, even though K has strictly larger m-th intrinsic volumes (i.e. V_m(K) > V_m(L)) for all m > i. It is then shown that, for each i = 1, ..., n, there is a class of bodies C{n,i}, called i-cylinder bodies of R^n, such that, if the body L with i-dimensional covering shadows is an i-cylinder body, then K will have smaller n-volume than L. The families C{n,i} are shown to form a strictly …
Using previous results about shadow systems and Steiner symmetrization, we prove that the local maximizers of the volume product of convex bodies are actually the global maximizers, that is: ellipsoids.
We consider the question how well a floating body can be approximated by the polar of the illumination body of the polar. We establish precise convergence results in the case of centrally symmetric polytopes. This leads to a new affine invariant which is related to the cone measure of the polytope.
We find general geometric conditions on a convex body of revolution K, in dimensions four and six, so that its intersection body IK is not a polar zonoid. We exhibit several examples of intersection bodies which are are not polar zonoids.
We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in John's position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in John's position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.
Define a body A to be able to hide behind a body B if the orthogonal projection of B contains a translation of the corresponding orthogonal projection of A in every direction. In two dimensions, it is easy to observe that there exist two objects such that one can hide behind another and have a larger area than the other. It was recently shown that similar examples exist in higher dimensions as well. However, the highest possible volume ratio for such bodies is still undetermined. We investigated two three-dimensional examples, one involving a tetrahedron and a ball and the other involving a te…
Classifications of $\rm{SL}(n)$ covariant function-valued valuations are established with some assumptions of continuity. New valuations, for example, weighted moment functions, are introduced and our classifications give unified characterizations of the Laplace transform on convex bodies, $L_p$ moment bodies, $L_p$ difference bodies, and polar $L_p$ moment bodies ($L_{-p}$ intersection bodies). Using the new classifications, we also establish some Euler-type relations.
In this paper, we propose and study the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure $μ$ and a continuous function $φ:(0,\infty)\rightarrow(0,\infty)$, there exists a convex body $K\in\mathcal{K}_0$ such that $K$ is an optimizer of the following optimization problems: \begin{equation*} \inf/\sup \bigg\{\int_{S^{n-1}}φ\big( h_L \big) \,d μ: L \in \mathcal{K}_{0} \ \text{and}\ |L^\circ|=ω_{n}\bigg\}. \end{equation*} The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on $φ,$ …
We will prove a reverse Rogers-Shephard inequality for log-concave functions. In some particular cases, the method used for general log-concave functions can be slightly improved, allowing us to prove volume estimates for polars of $\ell_p$-diferences of convex bodies whose polar bodies under some condition on the barycenter of their polar bodies.
Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies.
This paper is dedicated to study the sine version of polar bodies and establish the $L_p$-sine Blaschke-Santaló inequality for the $L_p$-sine centroid body. The $L_p$-sine centroid body $Λ_p K$ for a star body $K\subset\mathbb{R}^n$ is a convex body based on the $L_p$-sine transform, and its associated Blaschke-Santaló inequality provides an upper bound for the volume of $Λ_p^{\circ}K$, the polar body of $Λ_p K$, in terms of the volume of $K$. Thus, this inequality can be viewed as the "sine cousin" of the $L_p$ Blaschke-Santaló inequality established by Lutwak and Zhang. As $p\rightarrow \inf…
Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in R^n. The corresponding inequality to the conjecture is sometimes called the the reverse Blaschke-Santalo inequality. The conjecture is known in dimension two and in several special cases. In the class of unconditional convex bodies, Saint Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, w…
This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar.
In this paper, a new proof of the following result is given: The product of the volumes of an origin symmetric convex bodies $K$ in R^2 and of its polar body is minimal if and only if $K$ is a parallelogram.
In this paper we study the $\Z/2$ action on real Grassmann manifolds $G_{n}(\R^{2n})$ and $\widetilde{G}_{n}(\R^{2n})$ given by taking (appropriately oriented) orthogonal complement. We completely evaluate the related $\Z/2$ Fadell--Husseini index utilizing a novel computation of the Stiefel--Whitney classes of the wreath product of a vector bundle. These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For $n=2^a (2 b+1)$, $k=2^{a+1}-1$, $C$ a convex body in $\R^{2n}$, and $k$ real valued functions $α_1,\ldots,α_k$ continuous on conve…
Let $K, D$ be $n$-dimensional convex bodes. Define the distance between $K$ and $D$ as $$ d(K,D) = \inf \{λ| T K \subset D+x \subset λ\cdot TK \}, $$ where the infimum is taken over all $x \in R^n$ and all invertible linear operators $T$. Assume that 0 is an interior point of $K$ and define $$ M(K) =\int_{S^{n-1}} \| ω\|_K d μ(ω), $$ where $μ$ is the uniform measure on the sphere. Let $K^{\circ}$ be the polar body of $K$. We use the difference body estimate to prove that $K$ can be embedded into $R^n$ so that $$ M(K) \cdot M(K^{\circ}) \le C n^{1/3} \log^a n $$ for some absolute constants $C$ …
Let $K \subset {\mathbb R}^2$ be an $o$-symmetric convex body, and $K^*$ its polar body. Then we have $|K|\cdot |K^*| \ge 8$, with equality if and only if $K$ is a parallelogram. ($| \cdot |$ denotes volume). If $K \subset {\mathbb R}^2$ is a convex body, with $o \in {\text{int}}\,K$, then $|K|\cdot |K^*| \ge 27/4$, with equality if and only if $K$ is a triangle and $o$ is its centroid. If $K \subset {\mathbb R}^2$ is a convex body, then we have $|K| \cdot |[(K-K)/2)]^* | \ge 6$, with equality if and only if $K$ is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and…
The spherical centroid body of a centrally-symmetric convex body in the Euclidean unit sphere is introduced. Two alternative definitions - one geometric, the other probabilistic in nature - are given and shown to lead to the same objects. The geometric approach is then used to establish a number of basic properties of spherical centroid bodies, while the probabilistic approach inspires the proof of a spherical analogue of the classical polar Busemann-Petty centroid inequality.
It is proved that the simplex is a strict local minimum for the volume product, P(K)=min(vol(K) vol(K^z)), K^z is the polar body of K with respect to z, the minimum is taken over z in the interior of K, in the Banach-Mazur space of n-dimensional (classes of ) convex bodies. Linear local stability in the neighborhood of the simplex is proved as well. The proof consists of an extension to the non-symmetric setting of methods that were recently introduced by Nazarov, Petrov, Ryabogin and Zvavitch, as well as proving results of independent interest, concerning stability of square order of volumes …
Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such Minkowski valuations and a generalization of the notion of centroid bodies. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of volume inequalities for inte…
We prove a new family of inequalities, which compare the integral of a geometric convolution of non-negative functions with the integrals of the original functions. For classical inf-convolution, this type of inequality is called the Prékopa-Leindler inequality, which, restricted to indicators of convex bodies, gives the classical Brunn-Minkowski inequality. The convolution we consider is a different one, which arises from the study of the polarity transform for functions. While inf-convolution arises as the pull back of usual addition of convex functions under the Legendre transform, our geom…
We investigate the weighted $L_p$ affine surface areas which appear in the recently established $L_p$ Steiner formula of the $L_p$ Brunn Minkowski theory. We show that they are valuations on the set of convex bodies and prove isoperimetric inequalities for them. We show that they are related to $f$ divergences of the cone measures of the convex body and its polar, namely the Kullback-Leibler divergence and the Rényi-divergence.
We obtain a new extension of Rogers-Shephard inequality providing an upper bound for the volume of the sum of two convex bodies $K$ and $L$. We also give lower bounds for the volume of the $k$-th limiting convolution body of two convex bodies $K$ and $L$. Special attention is paid to the $(n-1)$-th limiting convolution body, for which a sharp inequality, which is equality only when $K=-L$ is a simplex, is given. Since the $n$-th limiting convolution body of $K$ and $-K$ is the polar projection body of $K$, these inequalities can be viewed as an extension of Zhang's inequality.
In a work by Artstein-Avidan and Milman the concept of polarity is generalized from the class of convex bodies to the larger class of convex functions. While the only self-polar convex body is the Euclidean ball, it turns out that there are numerous self-polar convex functions. In this work we give a complete characterization of all rotationally invariant self-polar convex functions on R^n.
We study new classes of convex bodies and star bodies with unusual properties. First we define the class of reciprocal bodies, which may be viewed as convex bodies of the form "$1/K$". The map $K\mapsto K^\prime$ sending a body to its reciprocal is a duality on the class of reciprocal bodies, and we study its properties. To connect this new map with the classic polarity we use another construction, associating to each convex body $K$ a star body which we call its flower and denote by $K^\clubsuit$. The mapping $K\mapsto K^\clubsuit$ is a bijection between the class $\mathcal{K}_0^n$ of convex …
Given any polar pair of convex bodies we study its conjugate face maps and we characterize conjugate faces of non-exposed faces in terms of normal cones. The analysis is carried out using the positive hull operator which defines lattice isomorphisms linking three Galois connections. One of them assigns conjugate faces between the convex bodies. The second and third Galois connection is defined between the touching cones and the faces of each convex body separately. While the former is well-known, we introduce the latter in this article for any convex set in any finite dimension. We demonstrate…
We extend the classical LR characterization of chirotopes of finite planar families of points to chirotopes of finite planar families of pairwise disjoint convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a chirotope of finite planar families of pairwise disjoint convex bodies if and only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the set of 3-subsets of J is a chirotope of finite planar families of pairwise disjoint convex bodies. Our main tool is the polarity map, i.e., the map that assigns to a convex body the set of lines missing its interi…
Motivated by long-standing conjectures on the discretization of classical inequalities in the Geometry of Numbers, we investigate a new set of parameters, which we call \emph{packing minima}, associated to a convex body $K$ and a lattice $Λ$. These numbers interpolate between the successive minima of $K$ and the inverse of the successive minima of the polar body of $K$, and can be understood as packing counterparts to the covering minima of Kannan & Lovász (1988). As our main results, we prove sharp inequalities that relate the volume and the number of lattice points in $K$ to the sequence of …
