Differentiability of distance functions in normed linear spaces with uniformly Gateaux differentiable norm

Abstract

Consider a non-empty proper closed subset K of a Banach space $(X, \| \cdot \|)$ where the norm is uniformly Gateaux (uniformly Frechet) differentiable. Then the associated distance function d is guaranteed to be Gateaux differentiable on a dense subset D of X\K. Furthermore, Gateaux (Frechet) differentiability of the distance function at a point $x \in$ X\K can be characterised in terms of the weak* sequences ${d' (x_n)}$ where ${x_n}$ is a sequence in D converging to x.