No dimension independent core-sets for containment under homothetics

Abstract

Abstract. This paper deals with the containment problem under homothetics which has the minimal enclosing ball (MEB) problem as a prominent representative. We connect the problem to results in classic convex geometry and introduce a new series of radii, which we call core-radii. For the MEB problem, these radii have already been considered from a different point of view and sharp inequalities between them are known. In this paper sharp inequalities between core-radii for general containment under homothetics are obtained. Moreover, the presented inequalities are used to derive sharp upper bounds on the size of core-sets for containment under homothetics. In the MEB case, this yields a tight (dimension independent) bound for the size of such core-sets. In the general case, we show that there are core-sets of size linear in the dimension and that this bound stays sharp even if the container is required to be symmetric.

Is a complete, reduced set necessarily of constant width?

“…The following proposition characterizes an optimal containment between two sets by their touching points (cf. Theorem 2.3 in [6]).…”

Additive colourful Carathéodory type results with an application to radii

“…Very recently, special attention has been paid to the radii functionals measured with respect to an arbitrary C ∈ K n (cf. [6,7]). This motivated us to study the behavior of the outer radius R(·, C) of the sum of a finite amount of convex bodies.…”

The asymmetry of complete and constant width bodies in general normed spaces and the Jung constant

“…Hence, involving this asymmetry has not only theoretical interest but is useful in computations, too (cf. [13]). The question of finding the most asymmetric sets of constant width is a well known topic of study (see, e. g. [28] for the Minkowski asymmetry or [19,Theorem 56] for the asymmetry of Besicovitch).…”

“…and Bohnenblust [8] (for arbitrary Minkowski spaces) (4) R(K)/D(K) ≤ n n + 1 are famous, widely studied and have been object of many improvements and extensions (e. g. in [7,8,13,14,31,34]). Explicitely mentioning [9], in many classical works in convexity, significant parts are devoted to geometric inequalities among the basic radii (see [8,35,44,50]), generalizations ( [6,12,21,31,42]), and between radii and other functionals ( [5,10,32]).…”

“…In recent years both, Jung's inequality as well as the Minkowski asymmetry have drawn renewed attention. Jung's inequality has been generalized and improved in several ways (see, e. g., [7,13,31]). In [46] the Minkowski asymmetry has been used as a parameter for improving geometric inequalities and in [4,30] especially as a connection between a geometric inequality and its restricted version to symmetric sets.…”