A simple geometric proof of a theorem for starshaped unions of convex sets

Let F be a family of n+1 convex sets in R^d, each n of which have a point in common, such that F is starshaped. If either all members of F are closed or all members of F are open, then the intersection of F is nonempty. This result which strengthens a theorem by V.Klee follows from a topological theorem of C.D.Horvath and M.Lassonde. We present a simple geometric proof in the spirit of Klee''s proof. This immediately provides an alternative proof of a Helly type theorem due to M.Breen. An abstract vector space variant of the above result is given, too.