Smooth antiproximinal sets
Let \(S(X)\) be the unit sphere of the real Banach space \(X\). A nonempty closed convex set \(M\subsetneqq X\) is said to be smooth if the conditions \(x\in M\), \(f,g\in S(X^*)\), \(f(x)=\sup\{ f(y): y\in M\}\), \(g(x)=\sup\{f(y):y\in M\}\) imply that \(f=g\). The set \(M \subsetneqq X\) is said to be antiproximinal if no point \(x\in X\setminus A\) has a nearest point in \(M\). At the beginning the author shows that a Banach space contains a convex smooth compact set if and only if \(X\) is separable. Using this fact one obtains the main result: Let \(X\) be a separable Banach space. If \(X\) contains a nonempty convex closed bounded antiproximinal set then \(X\) necessarily contains a nonempty smooth convex closed bounded antiproximinal set. In particular \(C[0,1]\), \(c_0\), \(c\) contain a smooth bounded convex centrally symmetric antiproximinal body.
scientific article; zbMATH DE number 1262573