## 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 \(c_0\) or \(\ell _p\), \(1<p<\infty \), 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## Notes

## References

- 1.Bourbaki, N.: Topological Vector Spaces. Chapters 1–5. Translated from the French by H. G. Eggleston and S. Madan. Elements of Mathematics (Berlin). Springer, Berlin (1987)zbMATHGoogle Scholar
- 2.Day, M.: Normed Linear Spaces. Third Edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 21. Springer, Heidelberg (1973)Google Scholar
- 3.Dubins, L.E.: On extreme points of convex sets. J. Math. Anal. Appl.
**5**, 237–244 (1962)MathSciNetCrossRefzbMATHGoogle Scholar - 4.García-Pacheco, F.J.: Non-continuous linear functionals on topological vector spaces. Banach J. Math. Anal.
**2**(1), 11–15 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 5.García-Pacheco, F.J.: Isometric reflections in two dimensions and dual L \({}^1\)-structures. Bull. Korean Math. Soc.
**49**(6), 1275–1289 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 6.García-Pacheco, F.J.: Translations, norm-attaining functionals and elements of minimum norm. Rev. Un. Mat. Argent.
**54**(1), 69–82 (2013)MathSciNetzbMATHGoogle Scholar - 7.García-Pacheco, F.J.: Convex components and multi-slices in real topological vector spaces. Ann. Funct. Anal.
**6**(3), 73–86 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 8.García-Pacheco, F.J.: Advances on the Banach–Mazur conjecture for rotations. J. Nonlinear Convex Anal.
**16**(4), 761–765 (2015)MathSciNetzbMATHGoogle Scholar - 9.García-Pacheco, F.J.: Vertices, edges and facets of the unit ball. J. Convex Anal.
**26**(1), 105–116 (2019)MathSciNetzbMATHGoogle Scholar - 10.García-Pacheco, F. J.: Naranjo-Guerra, E.: Inner structure in real vector spaces. Georgian Math. J. (2018). https://doi.org/10.1515/gmj-2018-0048