Prox-regular sets and Legendre-Fenchel transform related to separation properties

Abstract

This paper is devoted to nonconvex/prox-regular separations of sets in Hilbert spaces. We introduce the Legendre-Fenchel r-conjugate of a prescribed function and r-quadratic support functionals and points of a given set, all associated to a positive constant r. By means of these concepts we obtain nonlinear functional separations for points and prox-regular sets. In addition to such functional separations, we also establish geometric separation results with balls for a prox-regular set and a strongly convex set.

KEYWORDS:

• Prox-regular set
• strongly convex set
• Legendre-Fenchel transform
• semiconvexity
• geometric separation

AMS subject classifications:

• 49J52
• 49J53
• 46N10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second 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.