Related papers: A solution to Bezdek's conjecture

An old conjecture states that among all simplices inscribed in the unit sphere the regular one has the maximal mean width. An equivalent formulation is that for any centered Gaussian vector $(\xi_1,\dots,\xi_n)$ satisfying $\mathbb…

A two-dimensional body, exhibiting a slight rotational movement, moves in a rarefied medium of particles which collide with it in a perfectly elastic way. In previously realized investigations by the first two authors, Plakhov & Gouveia…

The aperture angle alpha(x, Q) of a point x not in Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the…

We establish a lower bound for the surface area of a closed, convex hypersurface in Euclidean space in terms of its displacement under continuous maps. As a result, a hypothesized lower bound for the volume of a Riemannian $n$-sphere,…

The Betke-Henk-Wills conjecture provides an upper bound for the lattice point enumerator $G(K, \Lambda)$ of a convex body in terms of its successive minima. While the conjecture is established for orthogonal parallelotopes, its validity for…

Let $m>1$ be an integer, $B_m$ the set of all unit vectors of $\Bbb R^m$ pointing in the direction of a nonzero integer vector of the cube $[-1, 1]^m$. Denote by $s_m$ the radius of the largest ball contained in the convex hull of $B_m$. We…

A ball polyhedron is a finite intersection of congruent balls in $\mathbb{R}^3$. These shapes arise in various contexts in discrete and convex geometry. We focus on Reuleaux polyhedra, the subclass of ball polyhedra whose centers and…

We revisit an ingenious argument of K. Ball to provide sharp estimates for the volume of sections of a convex body in John's position. Our technique combines the geometric Brascamp-Lieb inequality with a generalised Parseval-type identity.…

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of…

Let $H_d$ denote the smallest integer $n$ such that for every convex body $K$ in $\Re^d$ there is a $0<\lambda < 1$ such that $K$ is covered by $n$ translates of $\lambda K$. In the book \emph{Research problems in discrete geometry.} by…

We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the $d$-dimensional Euclidean space. When the radius $r$ of the balls is large, this volume can be approximated by a polynomial of…

Complementing our previous results, we give a classification of all isometries (not necessarily surjective) of the metric space consisting of ball-bodies, endowed with the Hausdorff metric. "Ball bodies" are convex bodies which are…

Let K be a d-dimensional convex body, and let $K^{(n)}$ be the intersection of n halfspaces containing $K$ whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an…

We prove a generalization of the hyperplane inequality for intersection bodies, where volume is replaced by an arbitrary measure $\mu$ with even continuous density and sections are of arbitrary dimension $n-k,\ 1\le k <n.$ If $K$ is a…

The Gaussian Correlation Conjecture states that for any two symmetric, convex sets in n-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the…

In light of the log-Brunn-Minkowski conjecture, various attempts have been made to define the geometric mean of convex bodies. Many of these constructions are fairly complex and/or fail to satisfy some natural properties one would expect of…

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced…

In this short note, we show that the inradius of a convex body is comparable to its volume divided by its surface area. We also give a simple formula, in terms of its volume and inradius, that is comparable to the volume of its intersection…

We consider the problem of wrapping three-dimensional solid bodies with a given planar sheet of paper, where the paper may be folded or wrinkled but not stretched or torn. We propose a conjecture characterising the maximumvolume solid…

Extending results of Suss and Hadwiger (proved by them for the case of convex bodies and positive ratios), we show that compact (respectively, closed) convex sets in the Euclidean space of dimension n are homothetic provided for any given…