A Chebyshev set and its distance function

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