*Published Paper*

**Inserted:** 30 nov 2011

**Last Updated:** 25 jul 2017

**Journal:** Indiana Univ. Math. J.

**Volume:** 62

**Number:** 1

**Pages:** 45--89

**Year:** 2013

**Abstract:**

We study regularity properties enjoyed by a class of real-valued upper semicontinuous functions $f:R^d\to R$ whose hypograph satisfies a geometric property implying, for each point $P$ on the boundary of hypo$\,f$, the existence of a sort of (uniform) subquadratic tangent hypersurface whose intersection with hypo$\,f$ in a neighbourhood of $P$ reduces to $P$. This geometric property generalizes both the concepts of semiconcave functions and functions whose hypograph has positive reach in the sense of Federer; the associated class of functions arises in the study of regularity properties for the minimum time function of certain classes of nonlinear control systems and differential inclusions.

We prove that these functions share several regularity properties with semiconcave functions. In particular, they are locally $BV$ and differentiable a.e. Our approach consists in providing upper bounds for the dimension of the set of nondifferentiability points. Moreover, a finer classification of the singularities can be performed according to the dimension of the normal cone to the hypograph, thus generalizing a similar result proved by Federer for sets with positive reach. Techniques of nonsmooth analysis and geometric measure theory are used.

**Keywords:**
Exterior sphere condition, sets with positive reach, reduced boundary

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