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=∩{cl conv Sz: z∈D∩reg S}, where reg S denotes the set of regular points of S, D is a relatively open subset of S containing the set lnc S of local nonconvexity points of S, and Sz={s∈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 bounded 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 Sz in place of cl conv Sz. 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.