抽象凸最优反导数
Abstract convex optimal antiderivatives
摘要：
在经典凸分析的背景下研究了反导数族及其包络之后，我们现在将这些概念和结果推广并应用到抽象凸分析的背景下。给定关于c-次微分的部分数据，我们考虑所有符合给定数据的c-凸c-反导数的集合。在一定的假设下，这个集合不是空的，并且包含它的下包络和上包络。我们用明确的公式表示这些最优反导数。一些著名的函数实际上是最优c-凸c-反导数。在一个应用中，我们指出了c-单调映射的Fitzpatrick函数的一个自然极小性，即它是极小反导数.在另一个应用中，在度量空间中，约束Lipschitz扩张问题自然符合我们在这里讨论的凸性概念。最优Lipschitz扩张恰好是最优反导数。这种方法产生明确的公式，这些扩展，其中最特殊的情况下，恢复众所周知的扩展，由于麦克沙恩和惠特尼。
英文摘要：
Having studied families of antiderivatives and their envelopes in the setting of classical convex analysis, we now extend and apply these notions and results in settings of abstract convex analysis. Given partial data regarding a c-subdifferential, we consider the set of all c-convex c-antiderivatives that comply with the given data. Under a certain assumption, this set is not empty and contains both its lower and upper envelopes. We represent these optimal antiderivatives by explicit formulae. Some well known functions are, in fact, optimal c-convex c-antiderivatives. In one application, we point out a natural minimality property of the Fitzpatrick function of a c-monotone mapping, namely that it is a minimal antiderivative. In another application, in metric spaces, a constrained Lipschitz extension problem fits naturally the convexity notions we discuss here. It turns out that the optimal Lipschitz extensions are precisely the optimal antiderivatives. This approach yields explicit formulae for these extensions, the most particular case of which recovers the well known extensions due to McShane and Whitney.
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