Equivalence among various derivatives and subdifferentials of the distance function

For a nonempty closed set C in a normed linear space X with uniformly Gâteaux differentiable norm, it is shown that the distance function dC is strictly differentiable at x∈X⧹C iff it is regular at x iff its modified upper or lower Dini subdifferential at x is a singleton iff its upper or lower Dini subdifferential at x is nonempty iff its upper or lower Dini derivative at x is subadditive. Moreover if X is a Hilbert space, then dC is Fréchet differentiable at x∈X⧹C iff its Fréchet subdifferential at x is nonempty. Many characteristics of proximally smooth sets and convex closed sets in a Hilbert space are also given.