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arXiv · 2212.09954

Singularities of Fitzpatrick and convex functions

Dmitry Kramkov·Mihai Sîrbu
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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\'{\i}\v{c}ek [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 $f$.

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Dmitry Kramkov, Mihai Sîrbu. 2022-12-20. Singularities of Fitzpatrick and convex functions. https://arxiv.org/abs/2212.09954

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