Differentiability of distance function and the proximinal condition implying convexity

Abstract

We establish a necessary and sufficient condition for the differentiability of the distance function generated by a nonempty closed set K in a real normed linear space X under a proximinality condition on K. We do not assume the uniform differentiability constraints on the norm of the space as in Giles (Proc Am Math Soc 104: 458–464, 1988). Hence, our result advances that of Giles (Proc Am Math Soc 104: 458–464, 1988). We prove that the proximinal condition of Giles (Proc Am Math Soc 104: 458–464, 1988) is true for almost suns. The proximinal condition ensures convexity of an almost sun in some class of strongly smooth spaces under a differentiability condition of the distance function. A necessary and sufficient condition is obtained for the convexity of Chebyshev sets in Banach spaces with rotund dual.