Abstract

In a pseudo-Euclidean space with scalar product S(⋅,⋅)S(\cdot, \cdot), we show that the singularities of projections on SS-monotone sets and of the associated Fitzpatrick functions are covered by countable c−cc-c surfaces having positive normal vectors with respect to the SS-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 ff can be covered by a countable collection of c−cc-c surfaces. We show that the normal vectors to these surfaces are restricted to the cone generated by F−FF-F, where F:=cl range ∇fF:= {\rm cl}\,{\rm range}\,\nabla f, the closure of the range of the gradient of ff.

Author information

Contact details are reproduced from the original publication and may be historical.

Dmitry Kramkov

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, U.S.A.
and: University of Oxford

kramkov@cmu.edu

Keywords

  • Convexity
  • subdifferential
  • Fitzpatrick function
  • projection
  • pseudo-Euclidean space
  • normal vector
  • singularity

Mathematics Subject Classification

26B25, 26B05, 47H05, 52A20

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Suggested citation

D. Kramkov, M. Sîrbu. “Singularities of Fitzpatrick and Convex Functions.” Journal of Convex Analysis 31 (2024), No. 3, 827–846. https://doi.org/10.68381/jca31040

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.