Common supports as fixed points

A family S of sets in R^d is sundered if for each way of choosing a point from rd+1 members of S, the chosen points form the vertex-set of an (r–1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in R^d, and for each partition (S, S), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S from the members of S. This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.