Some regularity results for a class of upper semicontinuous functions

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 will 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.