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.

中文翻译：

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与分离属性相关的 Prox-regular 集和 Legendre-Fenchel 变换

本文致力于希尔伯特空间中集合的非凸/近似正则分离。我们介绍了给定函数的勒让德-芬切尔r共轭和r二次支持泛函和给定集合的点，所有这些都与一个正常数r相关联。通过这些概念，我们获得了点和近似正则集的非线性函数分离。除了这样的功能分离，我们还用球建立了几何分离结果，用于近似正则集和强凸集。