EXTERNAL TANGENTS AND CLOSEDNESS OF CONE PLUS SUBSPACE

[摘要] When X and Y are convex subsets of a topological vector space E, an external tangent of the ordered pair (X, Y) is defined as an open ray T that issues from a point of X boolean AND Y, is disjoint from X boolean OR Y, and is such that X intersects each open halfspace containing T. It is shown that if E is a separable normed space, C is a closed convex cone in E, and L is a line through the origin in E, then the vector sum C + L = {c + l: c is an element of C, l is an element of L} is closed if and only if the pair (C, L) does not admit any external tangent. When S is a subspace of finite dimension greater than 1, closedness of C + S is shown to be equivalent to the nonexistence of external tangents of a certain pair (C', L,), where L is a line through the origin and C' is a second closed convex cone constructed from (C, S). Questions about the closedness of sets of the form cone + subspace arise from Various optimization problems, from problems concerning the extension of positive linear functions, and from certain problems in matrix theory. (C) 1994 Academic Press, Inc.

[发布日期] 1994-12-01