Advances on Ricceri’s most famous conjecture

García-Pacheco, Francisco Javier; Hill, Justin R.

doi:10.2298/fil1504829g

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Quasi-absolutely Convex Sets1

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2015

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“…The previous remark arises the question whether every totally anti-proximinal convex set containing 0 is quasi-absolutely convex. In a very recent paper (see [8,Example 3.6]), the authors of this manuscript have found an example of a convex set containing 0 which is not quasi-absolutely convex. To finish this section, we will show that in finite dimensional normed spaces the bounded, convex sets containing 0 are always quasi-absolutely convex.…”

Section: Quasi-absolutely Convex Setsmentioning

confidence: 99%

A partial positive solution to a conjecture of Ricceri

García-Pacheco¹,

Hill²

2015

TMNA

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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.

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Section: Quasi-absolutely Convex Setsmentioning

confidence: 99%