Non-Linear Inner Structure of Topological Vector Spaces

Abstract

Inner structure appeared in the literature of topological vector spaces as a tool to characterize the extremal structure of convex sets. For instance, in recent years, inner structure has been used to provide a solution to The Faceless Problem and to characterize the finest locally convex vector topology on a real vector space. This manuscript goes one step further by settling the bases for studying the inner structure of non-convex sets. In first place, we observe that the well behaviour of the extremal structure of convex sets with respect to the inner structure does not transport to non-convex sets in the following sense: it has been already proved that if a face of a convex set intersects the inner points, then the face is the whole convex set; however, in the non-convex setting, we find an example of a non-convex set with a proper extremal subset that intersects the inner points. On the opposite, we prove that if a extremal subset of a non-necessarily convex set intersects the affine internal points, then the extremal subset coincides with the whole set. On the other hand, it was proved in the inner structure literature that isomorphisms of vector spaces and translations preserve the sets of inner points and outer points. In this manuscript, we show that in general, affine maps and convex maps do not preserve inner points. Finally, by making use of the inner structure, we find a simple proof of the fact that a convex and absorbing set is a neighborhood of 0 in the finest locally convex vector topology. In fact, we show that in a convex set with internal points, the subset of its inner points coincides with the subset of its internal points, which also coincides with its interior with respect to the finest locally convex vector topology.

Geometric Invariants of Surjective Isometries between Unit Spheres

“…We refer the reader to [30][31][32][33] for a wider perspective on these concepts. Inner structure was introduced for the first time in ( [30], Definition 1.2) for non-convex sets, although it appears implicitly in [34,35] for convex sets.…”

“…Finally, let us see that inn(C) = int S X (C). Indeed, we use again the fact that C − c is a convex set with non-empty interior in ker( f ), so we call on ( [33], Lemma 5( 6)) to conclude that inn(C − c) = int ker( f ) (C − c). Since translations preserve inner points ( [30], Proposition 1.3), we conclude that inn(C)…”

Compact convex sets free of inner points in infinite‐dimensional topological vector spaces

“…In this paper, we will only make use of the inner structure of convex sets. We refer the reader to [6,8,9,11] for a wider perspective on these concepts. In [11,Theorem 5.1], it is proved that every non-singleton convex subset of any finite-dimensional vector space has inner points.…”

“…In view of [11, Theorem 4.2(2)], $$C$$ is absorbing in $$Y$$. According to [6, Theorem 6], $$C$$ is a neighborhood of 0 in $$Y$$ endowed with the finest locally convex vector topology. By applying [6, Lemma 5(6)], we have that $$\mathrm{inn}(C)=\mathrm{int}_Y(C)$$.…”

“…This inner structure (inner points) has the particularity that it is independent of the ambient space. It is not an easy task, as it can be seen in [6,9,11], to find non-trivial convex sets free of inner points. It is even harder to find non-trivial closed convex sets lacking inner points.…”