Sets of locally maximal visibility and finite unions of starshaped sets

Let S = cl(intS), where intS is an open connected subset of some linear top ological space, such that S contains all triangular regions whose (relative ) boundaries lie in S. If some finite subset Tof S has locally maximal visi bility in S, then V-T = S. Hence S is a finite union of starshaped sets who se kernels are determined by T. An analogous result holds for S open. Moreo ver, counterexamples show that neither the requirement on triangular region s nor the restriction to a finite set T can be deleted.