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- Title
Differentiability properties for a class of non-convex functions.
- Authors
Colombo, Giovanni; Marigonda, Antonio
- Abstract
Closed sets K ⊂ $$\mathbb R^{n}$$ satisfying an external sphere condition with uniform radius (called ϕ-convexity or proximal smoothness) are considered. It is shown that for $$\mathcal H^{n-1}$$ -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 $$\mathcal H^{n-1}$$ -a.e. the (De Giorgi) external normal. Then lower semicontinuous functions f : $$\mathbb R^{n}\rightarrow \mathbb R\cup\{ +\infty\}$$ with ϕ-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 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.
- Subjects
MATHEMATICAL functions; NONCONVEX programming; BOUNDARY element methods; NONSMOOTH optimization; LIPSCHITZ spaces; FUNCTION spaces; MEASURE algebras
- Publication
Calculus of Variations & Partial Differential Equations, 2006, Vol 25, Issue 1, p1
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-005-0352-7