Abstract convex optimal antiderivatives
Abstract convex optimal antiderivatives
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抽象凸最优反导数
DOI：
10.1016/j.anihpc.2012.01.004
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
S. Reich
中科院分区：
文献类型：
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作者：
Sedi Bartz;S. Reich
在研究了经典凸分析中的不定积分族及其包络之后，我们现在将这些概念和结果推广应用到抽象凸分析中。给定关于c次微分的偏数据，我们考虑符合给定数据的所有c-凸c-不定积分的集合。在一定的假设下，这个集合不为空，并且同时包含下信封和上信封。我们用显式公式表示这些最优不定积分。事实上，一些众所周知的函数是最优c-凸c-不定积分。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，即它是一个极小不定积分。在另一个应用中，在度量空间中，约束Lipschitz扩展问题自然地符合我们这里讨论的凸性概念。结果证明最优利普希茨扩展就是最优不定积分。这种方法为这些扩展提供了明确的公式，其中最特殊的情况恢复了McShane和Whitney的众所周知的扩展。©2012 Elsevier Masson SAS。版权所有。
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.© 2012 Elsevier Masson SAS. All rights reserved.