  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
[16]. Hence, our result advances that of Giles [16]. We prove that the
proximinal condition of Giles [16] 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.
