B-convexity

Given a homeomorphism Phi : X --> R-n one can define on the topological space X a set operator through the formula Co-Phi(A) = Phi(-1)(Co(Phi(A)). Such a convexity on X has all the topological, geometric and algebraic properties of the usual convexity on R-n; up to a change of variable, it is a linear convexity. In the context of convex analysis and optimization theory such operators were considered by Avriel (1972) and Ben Tal (1977). We consider a sequence on homeomorphisms Phi(r) : R-n --> R-n and we study the abstract convexity which is associated to the limit, in the appropriate sense, of the sequence of set operators A --> Co-Phir, (A); we call the limit-convexity B-convexity. On R-+(n) one can loosely say that this B-convexity is obtained from the usual linear convexity through the formal substitution + --> max. We end this article with some simple applications to duality and "max-programming".