We say that a star body $K$ is completely symmetric if it has centroid at the origin and its symmetry group $G$ forces any ellipsoid whose symmetry group contains $G$, to be a ball. In this short note, we prove that if all central sections of a star body $L$ are completely symmetric, then $L$ has to be a ball. A special case of our result states that if all sections of $L$ are origin symmetric and 1-symmetric, then $L$ has to be a Euclidean ball. This answers a question from \cite{R2}. Our result is a consequence of a general theorem that we establish, stating that if the restrictions in almost all equators of a real function $f$ defined on the sphere, are isotropic functions, then $f$ is constant a.e. In the last section of this note, applications, improvements and related open problems are discussed and two additional open questions from \cite{R} and \cite{R2} are answered.}
Let $K$ and $L$ be two convex bodies in ${\mathbb R^5}$ with countably many diameters, such that their projections onto all $4$ dimensional subspaces containing one fixed diameter are directly congruent. We show that if these projections have no rotational symmetries, and the projections of $K,L$ on certain 3 dimensional subspaces have no symmetries, then $K=\pm L$ up to a translation. We also prove the corresponding result for sections of star bodies.
The purpose of this paper is to study the reflections of a convex body. In particular, we are interested in orthogonal reflections of its sections that can be extended to reflections of the whole body. For this reason, we need to study the case of a non-spherical ellipsoid, where a surprising structure arises (Section 2). These results allow us to give a new characterization of ellipsoids in terms of their reflections and, on the other hand, to prove a result deeply related to a conjecture due to K. Bezdek.
The purpose of this paper is to answer the following question: If all hyperplane sections through the origin of a convex body are "equal", is the convex body "equal" to the ball? The meaning of the notion "equal" will change in the course of this paper. Similarly, we are interested in the following problem: If all orthogonal projections of a convex body onto hyperplanes are "equal", is the convex body "equal" to the ball? Topology and convex geometry are deeply interrelated in the solution and understanding of these problems.
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.
We provide general estimates which compare the quermassintegrals of a convex body $K$ in ${\mathbb R}^n$ with the averages of the corresponding quermassintegrals of the $k$-codimensional sections of $K$ over $G_{n,n-k}$. An example is the inequality $$α_{n,k,j}\frac{W_j(K)}{|K|}\leq\int_{G_{n,n-k}}\frac{W_j(K\cap F)}{|K\cap F|}dν_{n,n-k}(F)\leq β_{n,k,j}\frac{W_j(K)}{|K|}$$ where the constants $α_{n,k,j}$ and $β_{n,k,j}$ depend only on $n,k$ and $j$, which holds true for any centrally symmetric convex body $K$ in ${\mathbb R}^n$ and any $0\leq j\leq n-k-1\leq n-1$. Using these estimates we obt…
We consider the problem of comparing the volumes of two star bodies in an even-dimensional euclidean space $\mathbb R^{2n} = \mathbb C^n$ by comparing their cross sectional areas along complex lines (special 2-dimensional real planes) through the origin. Under mild symmetry conditions on one of the bodies a Busemann-Petty type theorem holds. Quaternionic and Octonionic analogs also hold. The argument relies on integration in polar coordinates coupled with Jensen's inequality. Along the way we provide a criterion that detects which centered bodies are {\it circular}. i.e., stabilized by multipl…
Let $K\subset\mathbb S^{d-1}$ be a convex spherical body. Denote by $Δ(K)$ the distance between two random points in $K$ and denote by $σ(K)$ the length of a random chord of $K$. We explicitly express the distribution of $Δ(K)$ via the distribution of $σ(K)$. From this we find the density of distribution of $Δ(K)$ when $K$ is a spherical cap.
This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different volumetric results are considered in a more general setting, with volume replaced by an arbitrary measure. Surprisingly, such a general approach works for a number of volumetric results. In particular, we discuss the Busemann-Petty problem on sections of convex bodies for arbitrary measures and the slicing problem for arbitrary measures. We present generalizations of…
In this manuscript, we study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if convex bodies $K, L$ satisfy $|K|θ^{\perp}| \le |L \cap θ^{\perp}|$ for all $θ\in S^{n-1}$, then $|K| \le |L|$. Firstly, we study the reverse question: in particular, we show that if $K, L$ are origin-symmetric convex bodies in John's position with $|K \cap θ^{\perp}| \le |L|θ^{\perp}|$ for all $θ\in S^{n-1}$ then $|K| \le \sqrt{n}|L|$. The condition we consider is weaker than both the conditions $|K \cap θ^…
Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture {\mf for symmetric convex bodies}. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a celebrated result of Hadwiger and, as usual, can be proved using ideas from equivariant topology. The second is an inequality relating the product volume to areas of certain sections and their duals. We observe that these ideas give a large family of convex sets in every dimensi…
The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condition on the measure for this problem to have a solution. In this paper, we address Alexandrov's problem for convex bodies in the hyperbolic space H m+1. After defining the Gauss curvature measure of an arbitrary hyperbolic convex body, we completely solve Alexandrov's problem in t…
