An approach to a Ricceri's conjecture
Let \(X\) be a normed space. For \(x\in X\), let \(d(x;A) = \inf \{\| x -e \| : e \in A\}\). A subset \(A\) of \(X\) is said to be anti-proximinal if for all \(x\in X\backslash A\), \(d(x;A)\) is not attained. \(A\) is called totally anti-proximinal if this property holds true for any norm on \(X\). A normed space \(X\) is said to have the anti-proximinal property if every totally anti-proximinal set in \(X\) is not rare. The Ricceri conjecture states ``there exists a non-complete normed space with the anti-proximinal property''. The author proves a result which can be considered a positive result in the direction of solving the Ricceri conjecture.
scientific article; zbMATH DE number 6084118