ENVELOPES FOR SETS AND FUNCTIONS II: GENERALIZED POLARITY AND CONJUGACY

Let X, Y be two nonempty sets, Phi an extended real-valued bivariate coupling function on X x Y and Gamma a subset of X x Y. The present paper provides extensions to the well-known generalized Phi-conjugacy and Gamma-polarity of diverse results of our previous work [2] related to Phi-conjucacy and Lambda-polarity, where Lambda is a subset of a vector space E and phi is a function on E defining the particular coupling function (x, y) -> phi(x - y) on E x E. A particular attention is devoted to the conjugacy functions (resp. polarity sets) which are mutually generating. Finally, for a superadditive conjugacy function Phi, we obtain a full description of the class of Phi-envelopes.