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 semi-concave 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.