Differentiability of distance functions and a proximinal property inducing convexity

Abstract:

In a normed linear space $X$, consider a nonempty closed set $K$ which has the property that for some $r > 0$ there exists a set of points ${x_0} \in X\backslash K,d({x_0}K) > r$, which have closest points $p({x_0}) \in K$ and where the set of points ${x_0} - r(({x_0} - p({x_0}))/||{x_0} - p({x_0})||)$ is dense in $X\backslash K$. If the norm has sufficiently strong differentiability properties, then the distance function $d$ generated by $K$ has similar differentiability properties and it follows that, in some spaces, $K$ is convex.