Illuminating and covering convex bodies

TL;DR

This paper introduces new covering and illumination numbers for convex bodies using proper translates, compares them with existing concepts, and provides examples demonstrating their non-triviality and potential significance.

Contribution

It defines novel covering and illumination numbers based on proper translates, analyzes their properties, and compares them with classical measures in convex geometry.

Findings

New illumination numbers are non-trivial and interesting.

Comparison shows relationships with classical covering and illumination numbers.

Examples illustrate the potential applications of these new concepts.

Abstract

Covering numbers of convex bodies based on homothetical copies and related illumination numbers are well-known in combinatorial geometry and, for example, related to Hadwiger's famous covering problem. Similar numbers can be defined by using proper translates instead of homothets, and even more related concepts make sense. On these lines we introduce some new covering and illumination numbers of convex bodies, present their properties and compare them with each other as well as with already known numbers. Finally, some suggestive examples illustrate that these new illumination numbers are interesting and non-trivial.