On translative coverings of convex bodies

Abstract

We introduce and study $t$-coverings in $E^n$, i.e., arrangements of proper translates of a convex body $K \subset E^n$ sufficient to cover $K$. First, we investigate relations between $t$-coverings of the whole of $K$ and $t$-coverings of its boundary only. Refining the notion of $t$-covering in several ways, we then derive, particularly for centrally symmetric convex bodies and $n = 2$, theorems which are interesting for the geometry of normed planes. These statements are related to respective generalizations of Ti\c{t}eica's and Miquel's theorem as well as to notions like Voronoi regions. We also compare $t$-coverings with coverings in the spirit of Hadwiger, using smaller homothetical copies of $K$ instead of proper translates. This is done via a slight modification of Boltyanski's and Hadwiger's notion of illumination. Finally, we give upper bounds on the cardinalities of $t$-coverings.