A Chebyshev set and its distance function
Let \(K\) be a Chebyshev subset of a smooth Banach space \(B\). An old open question in Banach space geometry asks if \(K\) must be convex. The answer is not known even if \(B\) is a Hilbert space. There are many beautiful partial results. For example, compact Chebyshev subsets of a smooth Banach space are convex. Other partial results involve conditions on \(B\), and its dual space, \(B^*\), or geometric or topological properties of \(K\). This paper shows that \(K\) is convex if the distance function, \(d(x)=\min\{\|x-k\|: k \in K\}\), satisfies smoothness conditions such as Gâteaux differentiability or continuity. When stricter smoothness conditions are assumed for \(B\) or \(B^*\), less restrictive conditions are required for \(d\). The paper contains, for example, the following nice statement: Theorem. If the norms on \(B\) and \(B^*\) are Fréchet differentiable, then \(K\) is convex if and only if \(d\) is Fréchet differentiable.
scientific article; zbMATH DE number 1891512
scientific article