New advances towards a (positive) solution to Ricceri?s (most famous)
Conjecture are presented. One of these advances consists of showing that a
totally anti-proximinal absolutely convex subset of a vector space is
linearly open. We also prove that if every totally anti-proximinal convex
subset of a vector space is linearly open then Ricceri?s Conjecture holds
true. Finally we demonstrate that the concept of total anti-proximinality
does not make sense in the scope of pseudo-normed spaces.
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 [7] and provides a partial positive solution to a Ricceri's Conjectured posed in [9] 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.
Abstract. We study some geometric properties related to the set Π X ∶= {(x, x * ) ∈ S X × S X * ∶ x * (x) = } obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element (h, k) ∈ H ⊕ H to Π H for H a Hilbert space.
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