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The faceless problem
Description:  Internal points were introduced in the literature of topological vector spaces to characterize the nest locally convex vector topology. Later on, they have been generalized to the context of real vector spaces by means of the inner points. Inner points can be seen as the most opposite concept to the extreme points. In this context comes in the "faceless problem", that is, the informally posed open problem of characterizing those convex sets free of proper faces. We prove that a non-singleton convex subset of a real vector space is free of proper faces if and only if every point of the convex set is an inner point. As a consequence, we prove the following dichotomy theorem: a point of a convex set of a real vector space is either a extreme point of the set or an inner point of some face of the set. We also characterize the non-singleton closed convex subsets of a topological vector space free of proper faces, which turn out to be the linear manifolds. An application of this allows us to show that the only minimal faces of a linearly bounded closed convex subset of a topological vector space are the extreme points. As an application of the technique we develop to prove our results, we easily construct dense proper faces of convex sets and non-dense proper faces whose closure is not a face. Finally, some applications to the Banach-Mazur Conjecture will be discussed.
Date:  2017-11-03
Start Time:   14:30
Speaker:  F. J. García-Pacheco (Univ. Cadiz, Spain)
Institution:  Univ. Cadiz, Spain
Place:  Room 5.5
Research Groups: -Analysis
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