Differentiability properties for a class of non-convex functions

Closed sets $K⊂R^n$ satisfying an external sphere condition with uniform radius (called $ϕ$-convexity or proximal smoothness) are considered. It is shown that for $H^{n−1}$-a.e. $x ∈ ∂K$ 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 $∂K$ and the unit proximal normal equals $H^{n−1}$-a.e. the (De Giorgi) external normal. Then lower semicontinuous functions $f : R^n → R ∪ {+∞}$ with $ϕ$-convex epigraph are shown, among other results, to be locally BV 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 $δ(x) > 0$ such that $f$ is semiconvex on $B(x, δ(x))$. We remark that such functions are neither convex nor locally Lipschitz, in general. Methods of nonsmooth analysis and of geometric measure theory are used.