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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.

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