ENVELOPES FOR SETS AND FUNCTIONS: REGULARIZATION AND GENERALIZED CONJUGACY

Let X be a vector space and let φ : X → R ∪ { − ∞ , + ∞ } be an extended real‐valued function. For every function f : X → R ∪ { − ∞ , + ∞ } , let us define the φ ‐envelope of f byf φ ( x ) = sup    y ∈ X φ ( x − y )−⋅f ( y ) ,where−·denotes the lower subtraction in R ∪ { − ∞ , + ∞ } . The main purpose of this paper is to study in great detail the properties of the important generalized conjugation map f ↦ f φ . When the function φ is closed and convex, φ ‐envelopes can be expressed as Legendre–Fenchel conjugates. By particularizing with φ =( 1 / p λ ) ∥ · ∥ p , for λ > 0 and p ⩾ 1 , this allows us to derive new expressions of the Klee envelopes with index λ and power p . Links between φ ‐envelopes and Legendre–Fenchel conjugates are also explored when − φ is closed and convex. The case of Moreau envelopes is examined as a particular case. In addition to the φ ‐envelopes of functions, a parallel notion of envelope is introduced for subsets of X . Given subsets Λ , C ⊂ X , we define the Λ ‐envelope of C asC Λ = ⋂ x ∈ C( x + Λ ) . Connections between the transform C ↦ C Λand the aforestated φ ‐conjugation are investigated.