A better decomposition theorem for simply connected m-convex sets

Abstract

A set S in R d is said to be m-convex, m ⩾ 2, if and only if for every m points in S, at least one of the line segments determined by these points lies in S. For S a closed m-convex set in R 2, various decomposition theorems have been obtained to express S as a finite union of convex sets. However, the previous bounds may be lowered further, and we have the following result: In case S is simply connected, then S is a union of σ( m) or fewer convex sets, where σ(m) = [(m − N)(m − 3 2 ) + 3 2 ] . Moreover, this result induces an improved decomposition in the general case as well.