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Distance Function Associated to a Prox-regular set

  • SI: Geometry, Optimization and Convex Analysis
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Abstract

In this paper, we provide in a general Hilbert space new characterizations of uniform prox-regularity involving outside but sufficiently close points of considered sets. We show that the complement of a prox-regular set is nothing but the union of closed balls with common radius. We derive from this that the prox-regularity of a given closed set is equivalent to the semiconvexity property of its distance function. Various estimates involving the metric projection mapping to a prox-regular set are also established.

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Acknowledgements

The first author has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No 823731 CONMECH.

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Authors and Affiliations

  1. Laboratoire de Modélisation Pluridisciplinaire et Simulations, Université de Perpignan Via Domitia, 66860, Perpignan, France

    Florent Nacry

  2. Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, 34095, Montpellier, France

    Lionel Thibault

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  1. Florent Nacry
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  2. Lionel Thibault
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Correspondence to Florent Nacry.

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Nacry, F., Thibault, L. Distance Function Associated to a Prox-regular set. Set-Valued Var. Anal 30, 731–750 (2022). https://doi.org/10.1007/s11228-021-00616-x

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