On Translative Coverings of Convex Bodies

We introduce and study t-coverings in En, i.e., arrangements of proper translates of a convex body K ⊂ En 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 Tiţeica’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.