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 $ker_RS = ∩{cl affB_z^R : z ∈ regS}$, where $ker_RS$ is the R-kernel of S, $regS$ denotes the set of regular points of S and $B_z^R = {s∈ S : z is R-visible from s via S}$. Furthermore, it is shown for a closed connected nonconvex subset S of X that $ker_RS = ∩{convB_z^R: z∈ D}$, where D is a relatively open subset of S containing the set $IncS$ 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 $sphS$ of spherical points of S in place of $regS$, 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 $slncS$, the set of strong local nonconvexity points of S, it is shown that $∩ {affA_z^R○:z ∈ slncS}⊆ qker_{R^○}S$, where $qker_{R^○}S$ is the quasi-$R^○$-kernel of S and $A_z^R○ = {s ∈ clS : z \textrm{ is clearly }R^○-\textrm{visible from s via S}}$, 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.