LNC points for m-convex sets

Abstract

Let S be closed, m-convex subset of Rd, S locally a full d-dimensional, with Q the corresponding set of lnc points of S. If q is an essential lnc point of order k then for some neighborhood U of q, Q⋂U is expressible as a union of k or fewer (d−2)-dimensional manifolds, each containing q For S compact, if to every q∈Q there corresponds a k>0 such that q is an essential lnc point of order k then Q may be written as a finite union of (d−2)-manifolds.For q any lnc point of S and N a convex neighborhood of q, N⋂ bdry  S⊈Q That is, Q is nowhere dense in bdry S. Moreover, if conv(Q⋂N)⫅S then Q⋂N is not homeomorphic to a (d−1)-dimensional manifold.