On Visibility and Covering by Convex Sets

A set X IR d is n-convex if among any n its points there exist two such that the segment connecting them is contained in X. Perles and Shelah have shown that any closed (n + 1)-convex set in the plane is the union of at most n 6 convex sets. We improve their bound to 18n 3 , and show a lower bound of order (n 2). We also show that if X IR 2 is an n-convex set such that its complement has one-point path-connectivity components, < 1, then X is the union of O(n 4 + n 2) convex sets. Two other results on the n-convex sets are stated in the introduction (Corollary 1.2 and Proposition 1.4).