# Inner structure of convex sets

- Francisco Javier García Pacheco Director

Defence university: Universidad de Cádiz

Year of defence: 2018

- Jarno Talponen Chair
- Kenier Castillo Secretary
- Fernando León Saavedra Committee member

Type: Thesis

## Abstract

In this dissertation we first show that every real or complex infinite dimensional normed space has an unbounded absolutely convex and absorbing subset with empty interior. As a consequence, we obtain that a real normed space is finite dimensional if and only if every convex subset containing 0 whose linear span is the whole space has non-empty interior. Then we prove that in every real or complex separable normed space with dimension strictly greater than 1 there exists a balanced and absorbing subset with empty interior which is dense in the unit ball. Explicit constructions of the previous sets are given. However these constructions vary depending whether the normed space is separable or not. Therefore, we later provide a unique construction by means of a family of balanced and absorbing sets with empty interior in every normed space of dimension strictly greater than 1. On the other hand, recall that the internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology, which turns out to be Hausdorff. As a consequence, internal points generalize topological interior points. We go further in this sense and generalize the concept of internal point in real vector spaces by introducing what we call inner points. By means of the inner points, we can provide an intrinsic characterization of linear manifolds that was not possible by using internal points. We also characterize the infinite dimensional real vector spaces by relying on the inner points of convex sets. Finally, we prove that in convex sets containing internal points, the set of inner points coincide with the one of internal points. Finally, we propose a study of the supporting vectors of continuous linear projections on Banach spaces, that is, the unit vectors at which a projection attains its norm. We first characterize supporting vectors of general continuous linear operators between Banach spaces. Then we characterize M-projections in terms of supporting vectors. Finally, by means of the supporting vectors again, we provide a characterization of projections of norm 1 on strictly convex Banach spaces.