Representations of starshaped sets in normed linear spaces

If S is a closed connected nonconvex locally compact and bounded subset of a real normed linear space or a closed connected nonconvex and bounded subset of a real reflexive Banach space, then ker S = boolean AND{cl conv S-z: z is an element of D boolean AND reg S}. where reg S denotes the set of regular points: of S, D is a relatively open subset of S containing the set inc S of local nonconvexity points of S, and S-z = {s is an element of S:z is visible from s via S}. An analogous intersection formula, with the set sph S of spherical points of S in place of reg S is shown to hold for a closed connected nonconvex and hounded subset S of a real Banach space which is uniformly convex and uniformly smooth. if the assumption of boundedness of S is dropped, then in all specified settings the above representations hold with intersections of conv S-z in place of cl conv S-z. This strengthens and complements results of Borwein and Strojwas, Stavrakas, and the author. Finally, the possibility of generating similar intersection formulae in other configuration set-space is discussed. (C) 2000 Academic Press.