Convex bodies: the brunn–minkowski theory

Computational geometry

Introduction 1. Packaging and covering densities 2. The existence of reasonably dense packings 3. The existence of reasonably economical coverings 4. The existence of reasonably dense lattice packings 5. The existence of reasonably economical lattice coverings 6. Packings of simplices cannot be very dense 8. Coverings with spheres cannot be very economical Bibliography Index.

Packing and Covering. By C. A. Rogers. Pp. viii, 111. 30s. 1964. (Cambridge)

THIS is a thoroughly good book which deserves to run through many editions. It is not (as its title might suggest) a book on mathematical logic or philosophy: it deals not with the nature of mathematics but with its content. Its purpose is to show, not by general disquisitions but by concrete examples, drawn from almost every branch of pure mathematics, how mathematicians think and what they do. What is Mathematics? An Elementary Approach to Ideas and Methods. By Richard Courant and Herbert Robbins. Pp. xix + 521. (London, New York and Toronto: Oxford University Press, 1941.) 25s. net.

What is Mathematics

The survey contains results related to different aspects of polyhedral approximation of convex bodies and some adjacent problems.

Approximation of convex sets by polytopes

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.

Ball-Polyhedra

We study two notions One is that of spindle convexity A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra We find analogues of several results on convex polyhedral sets for ball-polyhedra

Let ${C \subset \mathbb{R}^n}$ be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either ${\emptyset}$ , or ${\mathbb{R}^n}$ . The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Caratheodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces.

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Ball and spindle convexity with respect to a convex body

In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the d-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with $d+2$ vertices in every dimension, and for polytopes with $d+3$ vertices in odd dimensions. For polytopes with $d+3$ vertices in even dimensions we give a partial solution.

Maximum volume polytopes inscribed in the unit sphere

Let $C\subset {\mathbb R}^n$ be a convex body. We introduce two notions of convexity associated to C. A set $K$ is $C$-ball convex if it is the intersection of translates of $C$, or it is either $\emptyset$, or ${\mathbb R}^n$. The $C$-ball convex hull of two points is called a $C$-spindle. $K$ is $C$-spindle convex if it contains the $C$-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to $C$-spindle convex and $C$-ball convex sets. We study separation properties and Carath\'eodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc $C$, which is the length of an arc of a translate of $C$, measured in the $C$-norm, that connects two points. Then we characterize those $n$-dimensional convex bodies $C$ for which every $C$-ball convex set is the $C$-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some $C$-ball convex sets, and diametrically maximal sets in $n$-dimensional Minkowski spaces.

/pdf/ball-and-spindle-convexity-with-respect-to-a-convex-body-5613ssa1z5.pdf

Zsolt Lángi

Papers

Ball-Polyhedra

Ball-Polyhedra

Ball and spindle convexity with respect to a convex body

Maximum volume polytopes inscribed in the unit sphere

Ball and Spindle Convexity with respect to a Convex Body