On a Covering Property of Convex Sets

Let [Kx, K2, ■ ■ ■ } be a class of compact convex subsets of euclidean M-space with the property that the set of their diameters is bounded. It is shown that the sets A, can be rearranged by the application of rigid motions so as to cover the total space if and only if the sum of the volumes of all the sets A, is infinite. Also, some statements regarding the densities of such coverings are proved. If {A,} = {Kx, K2, . . .} is a class of compact convex sets of euclidean ^-dimensional space R", we say that {A",} can cover R", or that {A",} permits a space covering, if there are rigid motions a, so that Rn c LifLioiKi. The volume of Kt will be denoted by v(Kt), and the diameter by d(Kt). In a recent report [1], G. D. Chakerian has discussed the problem of finding necessary and sufficient conditions on v(Kt) and d(K/) in order that a class {A,} permits a space covering. In the joint paper [2] we have proved, using an idea of G. T. Sallee, the special case n = 2 of the following theorem: Theorem 1. Let {A-,} (/ = I, 2, . . . ) be a class of compact convex subsets of R" with the property that for some constant M and for i = 1, 2, ... , (1) d(Kt) < M. Then {A,} can cover R" if and only if OO (2) £ «(*/)=«>• I1 In the present paper this theorem will be proved in full generality. First we formulate and prove a theorem regarding coverings of the unit cube by a finite number of intervals (Theorem 2). From this theorem it will not only be possible to deduce Theorem 1, but also a more precise statement involving the densities of such coverings (Corollary 1). It should be mentioned that there is a kind of a "dual" of Theorem 2 concerning packings of convex sets rather than coverings (see Kosihski [5]). Theorem 3 shows the possibility of space coverings of density 1 by rather general collections of intervals. As a consequence of this theorem we can deduce another statement (Corollary 2) that establishes, under stronger assumptions than those in Corollary 1, a smaller bound for the densities of coverings of R" by convex sets. Received by the editors September 27, 1975 and, in revised form, January 5, 1976. AMS (MOS) subject classifications (1970). Primary 52A45; Secondary 05B40, 52A20.