APPOLLONIUS REVISITED - SUPPORTING SPHERES FOR SUNDERED SYSTEMS

When C is a ball in R-d and S is the sphere partial derivative C, we s ay that S supports a convex body B if S intersects B and either B subs et of or equal to C (then S is afar support) or the interior of C is d isjoint from B (then S is a near support). The focus here is on common supports for a system a of d + 1 bodies in R-d such that for each way of selecting a point from each member of B, the selected points are a ffinely independent and hence form the vertex-set df a d-simplex. The main result asserts that if (B', B '') is an arbitrary partition of B, then there exists a unique Euclidean sphere that is simultaneously a near support for each member of B' and a far support for each member o f B ''.