A Chebyshev Set and its Distance Function Original Research Article

We prove that in a Banach space X with rotund dual X* a Chebyshev set C is convex iff the distance function dC is regular on X\C iff dC admits the strict and Gâteaux derivatives on X\C which are determined by the subdifferential ∂∣∣x−x̄∣∣ for each x∈X\C and x̄∈PC(x)≔\{c∈C:∣∣x−c∣∣=dC(x)}. If X is a reflexive Banach space with smooth and Kadec norm then C is convex iff it is weakly closed iff PC is continuous. If the norms of X and X* are Fréchet differentiable then C is convex iff dC is Fréchet differentiable on X\C. If also X has a uniformly Gâteaux differentiable norm then C is convex iff the Gâteaux (Fréchet) subdifferential ∂−dC(x) (∂FdC(x)) is nonempty on X\C.