ENVELOPES FOR SETS AND FUNCTIONS: REGULARIZATION AND GENERALIZED CONJUGACY

Let 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 by f 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.