The paper is devoted to the abstract \({\mathcal{H}}\)-convexity of functions (where \({\mathcal{H}}\) is a given set of elementary functions) and its realization in the cases when \({\mathcal{H}}\) is the space of Lipschitz functions or the set of Lipschitz concave functions. The notion of regular \({\mathcal{H}}\)-convex functions is introduced. These are functions representable as the upper envelopes of the set of their maximal (with respect to the pointwise order) \({\mathcal{H}}\)-minorants. As a generalization of the global subdifferential of a convex function, we introduce the set of maximal support \({\mathcal{H}}\)-minorants at a point and the set of lower \({\mathcal{H}}\)-support points. Using these tools, we formulate both a necessary condition and a sufficient one for global minima of nonsmooth functions. In the second part of the paper, the abstract notions of \({\mathcal{H}}\)-convexity are realized in the specific cases when functions are defined on a metric or normed space \(X\) and the set of elementary functions is the space \({\mathcal{L}}(X,{\mathbb{R}})\) of Lipschitz functions or the set \({\mathcal{L}}\widehat{C}(X,{\mathbb{R}})\) of Lipschitz concave functions, respectively. An important result of this part of the paper is the proof of the fact that, for a lower semicontinuous function lower bounded by a Lipschitz function, the set of lower \({\mathcal{L}}\)-support points and the set of lower \({\mathcal{L}}\widehat{C}\)-support points coincide and are dense in the effective domain of the function. These results extend the known Brøndsted–Rockafellar theorem on the existence of the subdifferential for convex lower semicontinuous functions to the wider class of lower semicontinuous functions and go back to the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set, which is one of the most important results of classical convex analysis.

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关于Lipschitz（凹）函数集的函数的抽象凸性

本文专门讨论函数的抽象\（{\ mathcal {H}} \）-凸性（其中\（{\ mathcal {H}} \}是一组给定的基本函数）及其在以下情况下的实现\（{\ mathcal {H}} \）是Lipschitz函数或Lipschitz凹函数集的空间。引入了常规\（{\ mathcal {H}} \）-凸函数的概念。这些函数可表示为其最大（相对于逐点顺序）\（{\ mathcal {H}} \）- minorants集合的上包络。作为凸函数的全局次微分的一般化，我们介绍了一个点上的最大支持\（{\ mathcal {H}} \）- minorants集合和一个下限集合\（{\ mathcal {H}} \） -支持点。使用这些工具，我们为非光滑函数的全局最小值制定了必要条件和充分条件。在本文的第二部分中，\（{\ mathcal {H}} \）-凸性的抽象概念是在特定情况下实现的，这些情况是在度量或范数空间 \（X \）以及基本函数是Lipschitz函数的空间\（{\ mathcal {L}}（X，{\ mathbb {R}}）\\）或集合\（{\ mathcal {L}} \ widehat {C}（X， Lipschitz凹函数的{\ mathbb {R}}）\）。本文这一部分的重要结果是证明以下事实：对于以Lipschitz函数为下界的下半连续函数，下界\（{\ mathcal {L}} \） -支持点和一组较低的\（{\ mathcal {L}} \ widehat {C} \） -支持点是一致的，并且在函数的有效域中密集。这些结果将关于凸下半连续函数的次微分存在性的已知Brøndsted-Rockafellar定理扩展到下半连续函数的较宽类，并返回到Bishop-Phelps定理，该定理关于闭合凸边界上的支撑点集，这是经典凸分析的最重要结果之一。