A simple proof of Borsuk's conjecture in three dimensions

1. Borsuk (l) made the following conjecture: Every bounded set of points in Euclidean n-space E can be represented as the union of n+l sets of smaller diameter. Hadwiger (2,3,4) proved Borsuk's conjecture assuming the additional condition that the surface of the set is sufficiently smooth. On the other hand, a number of simple proofs have been supplied in the two-dimensional case (see, for example, Gale (5), where a stronger result is proved), as well as a complicated proof in the three-dimensional case (Eggleston (6)). In this note a simple proof is given for the conjecture in E. The proof is based on the idea (used also by Gale in E) of finding a suitable universal covering set (Deckel, couvercle, see Bonnesen & Fenchel (7), p. 87) for sets of diameter 1 in E and partitioning the covering set in four parts, each of diameter less than 1.