On subsets of S^n whose (n + 1)-point subsets are contained in open hemispheres

Abstract

We investigate the nature of subsets of spheres which satisfy a tameness condition associated with the Bieri–Groves FPm-conjecture. We find that there is a natural polyhedrality in the case of n-tame subsets of an (n − 1)-sphere. In the case n = 3 we establish a strong polyhedrality condition for certain maximal open 3-tame sets. Many examples are included