On the covering index of convex bodies

TL;DR

This paper introduces the covering index and weak covering index as new measures of how efficiently convex bodies can be covered by homothets, providing bounds, properties, and exact formulas for these indices.

Contribution

It defines the covering index and weak covering index, explores their properties, and establishes their behavior under geometric operations, advancing understanding of convex body coverings.

Findings

Covering index is lower semicontinuous on the Banach-Mazur space.

Affine d-cubes minimize the covering index in any dimension.

Circular disks maximize the covering index in the plane.

Abstract

Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce two relatives of the covering parameter called covering index and weak covering index, which upper bound well-studied quantities like the illumination number, the illumination parameter and the covering parameter of a convex body. Intuitively, the two indices measure how well a convex body can be covered by a relatively small number of homothets having the same relatively small homothety ratio. We show that the covering index is a lower semicontinuous functional on the Banach-Mazur space of convex bodies. We further show that the affine d-cubes minimize covering index in any dimension d, while circular disks maximize it in the plane. Furthermore, the covering index satisfies a nice compatibility with the operations of direct vector sum and vector sum. In fact, we obtain an exact formula for the covering index of a direct vector sum of convex bodies that works in infinitely many instances. This together with a minimization property can be used to determine the covering index of infinitely many convex bodies. As the name suggests, the weak covering index loses some of the important properties of the covering index. Finally, we obtain upper bounds on the covering and weak covering index.