Singularities of Fitzpatrick and Convex Functions

In a pseudo-Euclidean space with scalar product

, we show that the singularities of projections on

S S

-monotone sets and of the associated Fitzpatrick functions are covered by countable

c-c c − c

surfaces having positive normal vectors with respect to the

S S

-product. By L. Zajíček [On the differentiation of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29/104 (1979) 340–348], the singularities of a convex function

f f

can be covered by a countable collection of

c-c c − c

surfaces. We show that the normal vectors to these surfaces are restricted to the cone generated by

F-F F − F

, where

F:= {\rm cl}\,{\rm range}\,\nabla f F : = c l   r a n g e   ∇ f

, the closure of the range of the gradient of

f f

.