Some properties of semiconcave functions with general modulus

The class of semiconcave functions represents a useful generalization of the one of concave functions. Such an extension can be achieved requiring that a function satisfies a suitable one-sided estimate. In this paper, the structure of the set of points at which a semiconcave function fails to be differentiable—the singular set—is studied. First, we prove some results on the existence of arcs contained on the singular set. Then, we show how these abstract results apply to semiconcave solutions of Hamilton–Jacobi equations.  2002 Elsevier Science (USA). All rights reserved.