International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 3, Pages 513-528
doi:10.1155/S0161171281000379
LNC points for m-convex sets
Department of Mathematics, The University of Oklahoma, Norman 73019, Oklahoma, USA
Received 8 April 1980; Revised 4 September 1980
Copyright © 1981 Marilyn Breen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.