Related papers: On the maximum volume solid wrappable by a given s…
In this paper we provide a geometric condition satisfied by certain closed subsets of the Riemann sphere which implies that their hyperbolic convex hulls in $\mathbb{H}^3$ have infinite volume. As a corollary, we characterize continua in…
We present filling as a type of spatial subdivision problem similar to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In…
We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packing $\cal P$ with congruent…
We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean $3$-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove…
Let $K$ be a convex body in $\mathbb{R}^d$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a…
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…
In this article we prove a conjecture of Bezdek, Brass, and Harborth concerning the maximum volume of the convex hull of any facet-to-facet connected system of n unit hypercubes in the d-dimensional Euclidean space. For d=2 we enumerate the…
Using combinatorial methods, we derive several formulas for the volume of convex bodies obtained by intersecting a unit hypercube with a halfspace, or with a hyperplane of codimension 1, or with a flat defined by two parallel hyperplanes.…
In this note we investigate the behavior of the volume that the convex hull of two congruent and intersecting simplices in Euclidean $n$-space can have. We prove some useful equalities and inequalities on this volume. For the regular…
This paper provides the currently best known upper bound on the density of a packing in three-dimensional Euclidean space of two types of spheres whose size ratio is the largest one that allows the insertion of a small sphere in each…
If the n-dimensional unit sphere is covered by finitely many spherically convex bodies, then the sum of the inradii of these bodies is at least {\pi}. This bound is sharp, and the equality case is characterized.
Given a sphere of any radius $r$ in an $n$-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average…
One of the basic problems in discrete geometry is to determine the most efficient packing of congruent replicas of a given convex set $K$ in the plane or in space. The most commonly used measure of efficiency is density. Several types of…
In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an…
We study the problem of discrete geometric packing. Here, given weighted regions (say in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity.…
We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds…
We present a method to obtain upper bounds on covering numbers. As applications of this method, we reprove and generalize results of Rogers on economically covering Euclidean $n$-space with translates of a convex body, or more generally,…
A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most \pi. We can thus talk about the convexity of a set of points in terms of the…
Hadwiger's covering conjecture is that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller positive homothetic copies, with $2^n$ copies required only for affine images of $n$-cube. Convex hull of a ball and an…
Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth…