Differentiability properties for a class of non-convex functions

Closed sets K subset of R-n satisfying an external sphere condition with uniform radius (called (p-convexity or proximal smoothness) are considered. It is shown that for Hn-1-a.e. x epsilon partial derivative K the proximal normal cone to K at x has dimension one. Moreover if K is the closure of an open set satisfying a (sharp) nondegeneracy condition, then the De Giorgi reduced boundary is equivalent to a K and the unit proximal normal equals Hn-1-a.e. the (De Giorgi) external normal. Then lower semicontinuous functions f : R-n -> R U {+infinity} with phi-convex epigraph are shown, among other results, to be locally B V and twice L-n-a.e. differentiable; furthermore, the lower dimensional rectifiability of the singular set where f is not differentiable is studied. Finally we show that for L-n-a.e. x there exists delta(x) > 0 such that f is semiconvex on B(x, delta(x)). We remark that such functions are neither convex nor locally Lipschitz, in general. Methods of nonsmooth analysis and of geometric measure theory are used.