https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01514170Cabot, AlexandreAlexandreCabotIMB - Institut de Mathématiques de Bourgogne [Dijon] - UB - Université de Bourgogne - UBFC - Université Bourgogne Franche-Comté [COMUE] - CNRS - Centre National de la Recherche ScientifiqueJourani, AbderrahimAbderrahimJouraniIMB - Institut de Mathématiques de Bourgogne [Dijon] - UB - Université de Bourgogne - UBFC - Université Bourgogne Franche-Comté [COMUE] - CNRS - Centre National de la Recherche ScientifiqueThibault, LionelLionelThibaultIMAG - Institut Montpelliérain Alexander Grothendieck - UM - Université de Montpellier - CNRS - Centre National de la Recherche ScientifiqueCMM - Center for Mathematical Modelling - Centro de Modelamiento Matematico [Santiago] - UCHILE - Universidad de Chile = University of Chile [Santiago] - CNRS - Centre National de la Recherche ScientifiqueEnvelopes for sets and functions: regularization and generalized conjugacyHAL CCSD2017Convex-FunctionsWeak ConvexityDeconvolutionConvolutionPolaritiesDualityFormula[MATH] Mathematics [math]université de Bourgogne, UB_DRIVE2017-04-25 17:47:342022-12-11 09:11:442017-04-25 17:47:34enJournal articles10.1112/S00255793160003091Let X be a vector space and let phi : X -> R boolean OR {-infinity; +infinity} be an extended real-valued function. For every function f : X X -> R boolean OR {-infinity; +infinity}, let us define the phi-envelope of f byf phi(x) = sup(y is an element of X) phi(x - y) divided by f(y),where divided by denotes the lower subtraction in R boolean OR {-infinity, +infinity}. The main purpose of this paper is to study in great detail the properties of the important generalized conjugation map f -> f(phi). When the function ' is closed and convex, phi-envelopes can be expressed as Legendre-Fenchel conjugates. By particularizing with phi = (1/p lambda)parallel to.parallel to(p), for lambda > 0 and p >= 1, this allows us to derive new expressions of the Klee envelopes with index lambda and power p. Links between phi-envelopes and Legendre-Fenchel conjugates are also explored when -phi is closed and convex. The case of Moreau envelopes is examined as a particular case. In addition to the phi-envelopes of functions, a parallel notion of envelope is introduced for subsets of X. Given subsets Lambda, C boolean OR X, we define the 3 -envelope of C as C-Lambda = boolean AND(x is an element of C) (x + Lambda). Connections between the transform C -> C-Lambda and the aforestated phi-conjugation are investigated.