Singularities of Fitzpatrick and convex functions

Abstract

In a pseudo-Euclidean space with scalar product $S(\cdot ,\cdot )$, we show that the singularities of projections on $S$-monotone sets and of the associated Fitzpatrick functions are covered by countable $c-c$ surfaces having positive normal vectors with respect to the $S$-product. By Zaj\'ıček [24], the singularities of a convex function $f$ can be covered by a countable collection of $c-c$ surfaces. We show that the normal vectors to these surfaces are restricted to the cone generated by $F-F$, where $F:=\text{cl range }\nabla f$, the closure of the range of the gradient of $f$.

27 pages, to appear in Journal of Convex Analysis