Journal of Convex Analysis 31 (2024), No. 3, 827--846
Copyright Heldermann Verlag 2024
Singularities of Fitzpatrick and Convex Functions
Dmitry Kramkov
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, U.S.A.
kramkov@cmu.edu
Mihai Sîrbu
Department of Mathematics, The University of Texas, Austin, U.S.A.
sirbu@math.utexas.edu
[Abstract-pdf]
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 L.\,Zaj\'{\i}\v{c}ek [{\it 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 $f$ 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 $F := {\rm cl}\,{\rm range}\,\nabla f$, the closure of the range of the gradient of $f$.
Keywords: Convexity, subdifferential, Fitzpatrick function, projection, pseudo-Euclidean space, normal vector, singularity.
MSC: 26B25, 26B05, 47H05, 52A20.
[ Fulltext-pdf (163 KB)] for subscribers only.