Differentiability properties for a class of non-convex functions

Abstract

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

Quantitative isoperimetric inequalities for a class of nonconvex sets

“…Consequently (see [9,Theorem 4.3]), the reduced boundary coincides H n−1 -a.e. with the topological boundary.…”

“…We deal mainly with results on the line of [17] for a class of nonconvex sets called ϕ-convex sets. The concept of ϕ-convexity appears, for example, in connection with curvature measures (see [15]), with control theory (see [9,10]), or with variational problems (see [5,6]), and shares several properties with convex set. ϕ-convex sets are defined (see Definition 2.5 below) through a suitable external sphere condition with locally uniform radius, proportional to 1/ϕ.…”

Variational Analysis and Regularity of the Minimum Time Function for Differential Inclusions

“…Denote by L n and H d the Lebesgue n-dimensional measure and the Hausdorff d-dimensional measure, respectively. We recall here some regularity properties of functions whose epigraph is ϕ-convex (see [16]). …”

Hypographs satisfying an external sphere condition and the regularity of the minimum time function

“…In [9], G. Colombo and A. Marigonda proved that merely lower semicontinuous functions whose hypograph/epigraph has positive reach still enjoy some regularity properties of semiconcave/semiconvex functions, including twice a.e. differentiability, yet not being locally Lipschitz (see Theorem 2.1 below).…”