2024/09/06 by Anton Schiela, Roland Herzog, Schiela, Anton +3 · 2 citations
Engineering · #Advanced Fiber Optic Sensors #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2409.04492
openalex publication_date 2024/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.