An approach to a Ricceriʼs Conjecture

A totally anti-proximinal subset of a vector space is a non-empty proper subset which does not have a nearest point whatever is the norm that the vector space is endowed with. A Hausdorff locally convex topological vector space is said to have the (weak) anti-proximinal property if every totally anti-proximinal (absolutely) convex subset is not rare. A Ricceriʼs Conjecture posed in Ricceri (2007) [5] establishes the existence of a non-complete normed space satisfying the anti-proximinal property. In this manuscript we approach this conjecture in the positive by proving that a Hausdorff locally convex topological vector space has the weak anti-proximinal property if and only if it is barrelled. As a consequence, we show the existence of non-complete normed spaces satisfying the weak anti-proximinal property.