非线性 Fenchel 共轭
Nonlinear Fenchel Conjugates
AI总结：
本文将经典 Fenchel 共轭从线性空间推广到任意集合，用非线性测试函数替代线性测试函数，建立双共轭等性质，并在流形与 Lie 群上获得次微分、下确界卷积及凸性结果。
摘要：
经典的 Fenchel 共轭概念是为定义在线性空间上的扩充实值函数而量身定制的。在本文中，我们将这一概念推广到定义在任意集合上的函数，这些集合完全不必具有任何结构。这一推广是通过用一般的非线性测试函数替代线性测试函数实现的。因此，我们将其称为非线性 Fenchel 共轭。我们研究了包括 Fenchel–Moreau 双共轭定理在内的基本性质。当定义域具有额外结构时，将测试函数限制在适当子集上可以导出进一步结果。例如，在光滑流形上，限制为光滑测试函数使我们能够陈述关于黏性 Fréchet 次微分的 Fenchel–Young 定理。在 Lie 群上，限制为实值群同态将非线性 Fenchel 共轭与下确界卷积联系起来，并产生一种凸性概念。
英文摘要：
The classical concept of Fenchel conjugation is tailored to extended real-valued functions defined on linear spaces. In this paper we generalize this concept to functions defined on arbitrary sets that do not necessarily bear any structure at all. This generalization is obtained by replacing linear test functions by general nonlinear ones. Thus, we refer to it as nonlinear Fenchel conjugation. We investigate elementary properties including the Fenchel-Moreau biconjugation theorem. Whenever the domain exhibits additional structure, the restriction to a suitable subset of test functions allows further results to be derived. For example, on smooth manifolds, the restriction to smooth test functions allows us to state the Fenchel-Young theorem for the viscosity Fréchet subdifferential. On Lie groups, the restriction to real-valued group homomorphisms relates nonlinear Fenchel conjugation to infimal convolution and yields a notion of convexity.
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