Cylindrical Partitions of Convex Bodies

Summary

A cylindrical partition of a convex body in ℝⁿ is a partition of the body into subsets of smaller diameter, obtained by intersecting the body with a collection of mutually parallel convex-base cylinders. Convex bodies of constant width are characterized as those that do not admit a cylindrical partition. The main result is a finite upper bound, exponential in n, on the minimum number b_c(n)of pieces needed in a cylindrical partition of every convex body of nonconstant width in Rⁿ. (A lower bound on b_c(n), exponential in, is a consequence of the construction of Kalai and Kahn for counterexamples to Borsuk's conjecture.) We also consider cylindrical partitions of centrally symmetric bodies and of bodies with smooth boundaries.

1. Introduction and Preliminaries

Throughout this article, M denotes a compact subset of ℝⁿ containing at least two points. By diamM we denote the maximum distance between points of M, but diameter of M also means the line segment connecting any pair of points of M that realize this distance (ambiguity is always avoided by the context). A Borsuk partition of M is a family of subsets of M, each of diameter smaller than diamM, whose union contains M. The Borsuk partition number of M, denoted by b(M), is the minimum number of sets needed in a Borsuk partition of M. It is obvious that b(M) is finite. It is also obvious that the maximum of b(M) over all bounded sets M in ℝⁿ exists and is bounded above exponentially in n, since every set of diameter d is contained in a ball of radius d.