Generalized subdifferential of the distance function

Abstract:

We derive the proximal normal formula for almost proximinal sets in a smooth and locally uniformly convex Banach space. Our technique leads us to show the generic Fréchet smoothness of the distance function in the case the norm is Fréchet smooth, and we derive a necessary and sufficient condition for the convexity of a Chebyshev set in a Banach space $X$ with norms on $X$ and $X^*$ locally uniformly convex.