Singularities of Fitzpatrick and Convex Functions
Journal of convex analysis, Tome 31 (2024) no. 3, pp. 827-846
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 $f$.
Classification :
26B25, 26B05, 47H05, 52A20
Mots-clés : Convexity, subdifferential, Fitzpatrick function, projection, pseudo-Euclidean space, normal vector, singularity
Mots-clés : Convexity, subdifferential, Fitzpatrick function, projection, pseudo-Euclidean space, normal vector, singularity
@article{JCA_2024_31_3_JCA_2024_31_3_a5,
author = {D. Kramkov and M. S{\^\i}rbu},
title = {Singularities of {Fitzpatrick} and {Convex} {Functions}},
journal = {Journal of convex analysis},
pages = {827--846},
year = {2024},
volume = {31},
number = {3},
url = {http://geodesic.mathdoc.fr/item/JCA_2024_31_3_JCA_2024_31_3_a5/}
}
D. Kramkov; M. Sîrbu. Singularities of Fitzpatrick and Convex Functions. Journal of convex analysis, Tome 31 (2024) no. 3, pp. 827-846. http://geodesic.mathdoc.fr/item/JCA_2024_31_3_JCA_2024_31_3_a5/