Conjugacies adapted to lower semicontinuous functions

We revisit a remarkable duality devoted to lower semicontinuous functions. We compare its definition in terms of a coupling with its definition in terms of linear-like (or elementary) functions. We consider several variants. Then, we deal with the passage from smoothness to rotundity and the reverse passage and we examine the transfer of boundedness properties to coercivity properties. The relevance of this conjugacy for mathematical programming problems is pointed out.