In this paper, we present diverse new metric properties that prox-regular sets shared with convex ones. At the heart of our work lie the Legendre-Fenchel transform and complements of balls. First, we show that a connected prox-regular set is completely determined by the Legendre-Fenchel transform of a suitable perturbation of its indicator function. Then, we prove that such a function is also the right tool to extend, to the context of prox-regular sets, the famous connection between the distance function and the support function of a convex set. On the other hand, given a prox-regular set, we examine the intersection of complements of open balls containing the set. We establish that the distance of a point to a prox-regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point. This is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes. To the best of our knowledge, these results are new in the literature and show that the class of prox-regular sets have good properties known in convex analysis.

中文翻译：

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prox-regular 集合的新度量属性

在本文中，我们提出了与凸集共享的近似正则集的各种新度量属性。我们工作的核心是 Legendre-Fenchel 变换和球的补充。首先，我们证明了一个连通的正则集合完全由其指示函数的适当扰动的 Legendre-Fenchel 变换决定。然后，我们证明这样的函数也是将距离函数和凸集的支持函数之间著名的联系扩展到近似正则集的上下文的正确工具。另一方面，给定一个近似正则集合，我们检查包含该集合的开球的补集的交集。我们确定一个点到一个近似正则集的距离是该点与分隔该集和该点的所有此类补集的边界之间的最大距离。这与根据分离超平面表示与凸集的距离的已知结果一致。据我们所知，这些结果在文献中是新的，并且表明近似正则集的类别在凸分析中具有已知的良好特性。