No dimension independent Core-Sets for Containment under Homothetics

TL;DR

This paper explores containment problems under homothetics, introduces core-radii, and establishes sharp inequalities and bounds for core-set sizes, including a tight dimension-independent bound for the minimal enclosing ball case.

Contribution

It introduces core-radii and derives sharp inequalities, providing the first dimension-independent bounds for core-set sizes in the MEB problem and linear bounds for general cases.

Findings

Sharp inequalities between core-radii are established.

Dimension-independent bounds for core-set sizes in MEB are proven.

Core-sets of linear size in dimension exist for general containment under homothetics.

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.