On coverings of Euclidean space by convex sets

Abstract

Let j/ί = {KuK2, - - -} be an infinite countable class of compact convex subsets of euclidean n -dimensional space Rn. We shall say that % permits a space covering or, more precisely, a covering of Rn, if there are rigid motions σu σ2, such that Rn cU^i^Jζ. In this paper we concern ourselves with necessary and sufficient conditions in order that a given class 3ίf permits a space covering. If the set of diameters {d(Ki): K^ E K} is bounded, the problem has already been solved in [3] by showing that in this case JC permits a covering of Rn if and only if the series Σ7=ιv(Ki) of volumes v(Kt) diverges. (The same result holds obviously without any restrictions on the diameters if n = 1.) On the other hand, if {d(Ki)} is unbounded and n > 1, it is not difficult to see (cf. [1] and [2]) that the divergence of this series is no longer sufficient but only necessary. Only in the special case n = 2 are some necessary and sufficient conditions known [2]. Our principal results are stated in the following §2. Theorem 1 gives an inductive criterion that enables one to decide whether a given % permits a space covering. Theorems 2 and 3 serve the same purpose but are of a more explicit nature, involving the divergence of infinite series of geometric invariants associated with the members of 3ίf. Other results, regarding coverings by balls, boxes (i.e. isometric images of ndimensional intervals), and 2-dimensional sets, are stated and discussed in the same section. This is followed by the proofs of three lemmas in §3. Lemma 1 appears to be of some independent interest. §4 contains the proofs of our theorems.