We investigate the nature of subsets of spheres which satisfy a tameness condition associated with the Bieri-Groves FP_m-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.

The second author wishes to thank the Department of Mathematics at Cornell University, the Mittag-Leffler Institute, Djursholm, and also wishes to thank the Goethe-Universität Frankfurt for their generous support. The third author was supported in part by EPSRC grant EP/I033754/1.