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3887. Strongly convex sets with variable radii

Invited abstract in session TB-42: Variational Methods in Vector Optimization, stream Variational Analysis and Continuous Optimization.

Tuesday, 10:30-12:00
Room: 98 (building: 306)

Authors (first author is the speaker)

1. Florent Nacry
Université Perpignan

Abstract

This talk is devoted to the new class of strongly convex sets with variable radii in Hilbert spaces recently introduced by L. Thibault and the author. Such a strong convexity property can be seen as the convex counterpart of the famous phi-convexity introduced by A. Canino in 1988 and thoroughly developed in a 2010 survey by G. Colombo and L. Thibault under the name prox-regularity with variable thickness (also known in the case of constant radius/thickness as proximally smooth sets, positively reached sets, weakly convex sets, O(2)-convex sets, etc.).

Roughly speaking, a strongly convex set with a variable radius (is obviously convex and) has its curvature locally bounded from below by an appropriate function. Whenever the latter function is constant, our strong convexity coincides with the usual strong convexity of sets (see, e.g., a recent survey by V.V. Goncharov and G.E. Ivanov). Strongly convex sets have been involved in various mathematics, including: separation properties, minimal time problem, preservation of prox-regularity, linear differential games, sweeping processes, etc.

The main aim of the presentation is to provide numerous properties and characterizations of strongly convex sets with variable radii through the differentiability of the so-called farthest distance function, the Lipschitz behavior of the farthest point mapping and the strong monotonicity of truncated normals. Perspectives and open questions would be also provided.

Keywords

Status: accepted

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