A partial positive solution to a conjecture of Ricceri

Abstract

In this manuscript we introduce a new class of convex sets called quasi-absolutely convex and show that a Hausdorff locally convex topological vector space satisfies the weak anti-proximinal property if and only if every totally anti-proximinal quasi-absolutely convex subset is not rare. This improves results from [F.J. García-Pacheco, An approach to a Ricceri's conjecture, Top. Appl. 159 (2012), 3307-3313] and provides a partial positive solution to a Ricceri's Conjecture posed in [B. Ricceri, Topological problems in nonlinear and functional analysis, Open Problemsin Topology II (E. Pearl, ed.), Elsevier, 2007, 585-593] with many applications to the theory of partial differential equations. We also study the intrinsic structure of totally anti-proximinal convex subsets proving, among other things, that the absolutely convex hull of a linearly bounded totally anti-proximinal convex set must be finitely open. Finally, a new characterization of barrelledness in terms of comparison of norms is provided.