Geometric-combinatorial characteristics of cones

Abstract

It is shown for a proper closed locally compact subset S of a real normed linear space X that (Error rendering LaTeX formula) , where is the R-kernel of S, denotes the set of regular points of S and (Error rendering LaTeX formula) . Furthermore, it is shown for a closed connected nonconvex subset S of X that (Error rendering LaTeX formula) , where D is a relatively open subset of S containing the set of local nonconvexity points of S. If X is a uniformly convex and uniformly smooth real Banach space, then the first of these formulae is shown to hold with the set of spherical points of S in place of , and the second one for a closed connected nonconvex set S. For a connected subset S of a real topological linear space L with nonempty , the set of strong local nonconvexity points of S, it is shown that (Error rendering LaTeX formula) , where is the quasi- -kernel of S and (Error rendering LaTeX formula) , and that the equality holds provided, in addition, S is open. In conjunction with an infinite-dimensional version of Helly's theorem for flats, these intersection formulae generate Krasnosel‘skii-type characterizations of cones and quasi-cones. All this parallels the research done recently by the author for starshaped and quasi-starshaped sets.