A Solution to the Faceless Problem
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A Solution to the Faceless Problem F. J. García-Pacheco1 Received: 21 March 2019 © Mathematica Josephina, Inc. 2019
Abstract Internal points were introduced in the literature of topological vector spaces to characterize the finest 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 manuscript we solve the “faceless problem”, that is, the informally posed open problem of characterizing those convex sets free of proper faces. Indeed, 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 or an inner point of some face. 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 possible 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. This fact allows us to equivalently renorm any infinite dimensional Banach space so that its unit ball contains a non-closed face. Even more, we prove that every infinite dimensional Banach space containing an isomorphic copy of c0 or p , 1 < p < ∞, can be equivalently renormed so that its unit ball contains a face whose closure is not a face. Keywords Real vector space · Convex set · Face · Extreme point Mathematics Subject Classification Primary 15A03; Secondary 52A07 · 46A55
The author was supported by MTM2014-58984-P (this project has been funded by the Spanish Ministry of Economy and Competitivity and by the European Fund for Regional Development FEDER).
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F. J. García-Pacheco [email protected] Department of Mathematical Sciences, University of Cadiz, 11510 Puerto Real, Spain
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F. J. García-Pacheco
1 Introduction and Basic Notions All the vector spaces considered throughout this manuscript will be over the reals. The concept of internal point is proper of the theory of topological vector spaces and it is very useful to study the geometry of the convex bodies. A point of a subset of a vector space is said to be an internal point provided that every line passing through the point has a small interval around the point contained in the set. Internal points characterize the finest locally convex vector topology and also serve for stating most algebraic separation results. We refer the reader to [1, TVS II.26] for a wide perspective on internal points in topological vector spaces. Internal points are naturally generalized by the inner points. A point x of a convex set C of a vector space X is
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