On the Geometry of Locally Nonconical Convex Sets

Abstract

A convex subset Q of a Hausdorff topological vector space is called locally nonconical (LNC) if for every two points x,y∈Q there is a relative neighborhood U of x in Q such that U+\(\tfrac{1}{2}\)(y-x) ⊂Q. A geometric characterization (Theorem 2.2) of closed LNC sets with nonempty interior in a Hilbert space is supplied. It states that any proper line segment ]x,y[ contained in bd(Q), the topological boundary of Q, lies inside a relative neighborhood in bd(Q) composed of parallel line segments. It is shown that one half of this characterization, at least, generalizes to the setting of a locally convex Hausdorff topological vector space (LCHTVS). This leads to the observation that the set ext(Q) of extreme points of any LNC set Q in an LCHTVS is closed. Finally, it is proven that, in the same setting, all LNC sets are uniformly stable and, hence, stable.