On Nonconvex Version of the Inequality of Clarke and Ledyaev

The purpose of the paper is to extend the Clarke-Ledyaev multidirectional mean value inequality that estimates in terms of the subdifferential of a lower semicontinous function the difference between the infimum of the function on certain closed, bounded and convex set and its value on a certain point. The assumption of convexity is relaxed by showing that a similar inequality holds for any closed and bounded set and any point outside its closed convex hull from which the set “seems convex”. The boundaries of convex set seem convex from each point outside its closed convex hull. The technique applied allows proving the Clarke-Ledyaev inequality without assuming that the function is bounded bellow. Also, in the convex case the infimum can be taken only over the boundary of the set. An inverse theorem shows that for arbitrary lower semicontinuous function no inequality of such type can be expected if the set does not seem convex from the point. If it does seem convex then no stronger inequality can be obtained in general.