External Tangents and Closedness of Cone + Subspace

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 ∩ Y, is disjoint from X ∪ 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 ∈ C, ℓ ∈ 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.