An Optimal Krasnosel'skii-Type Theorem for the Dimension of the Kernel of a Starshaped Set
J. Cel · Bulletin of the London Mathematical Society · 1995
It is shown for a closed connected nonconvex and locally compact subset S of a real normed linear space that ker S = ∩ {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 set S in a real Banach space which is uniformly convex and uniformly smooth. A routine procedure then leads to a Krasnosel'skii-type characterization of the dimension of the kernel of a closed connected nonconvex subset S of Rd with lnc S bounded. This improves a recent result by removing the hypothesis of compactness of S and settles an open problem.