Supplemental subsets of Sn

Two angles are called supplementary when their sum is π. I propose that two non-empty subsets of Sn be called supplementary when no point of one is within 90 degrees of a point in the other and neither can be extended without violating the rule. Two supplementary subsets are each convex and are separated by an equator (which is an Sn−1). Two supplementary subsets of S1 subtend supplementary angles.

The supplementary subset of a polyhedral subset is polyhedral. Alas, the content of a subset does not determine the content of its supplement. This makes generalizations of Gauss-Bonnet more complex.

A subset of Sn is convex if it is the intersection of some non-empty set of hemispheres. The shorter geodesic path between two points in a convex set lies within that set.

This notion bears on curvatures at vertices of polyhedra. Here.