Differentiability properties for a class of non-convex functions

Closed sets $K\subset \mathbb R^{n}$ satisfying an external sphere condition with uniform radius (called $\varphi$-convexity, proximal smoothness, or positive reach) are considered. It is shown that for $\mathcal H^{n-1}$-a.e. $x\in \partial K$ 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 $\partial K$ and the unit proximal normal equals $\mathcal H^{n-1}$-a.e. the (De Giorgi) external normal. Then lower semicontinuous functions $f:\mathbb R^{n}\rightarrow \mathbb R\cup\{ +\infty\}$ with $\varphi$-convex epigraph are shown, among other results, to be locally $BV$ and twice $\mathcal L^{n}$-a.e. differentiable; furthermore, the lower dimensional rectifiability of the singular set where $f$ is not differentiable is studied. Finally we show that for $\mathcal L^{n}$-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.