Before revision:  We propose several characterizations of the (AIP ) condition and also study the relation with (NA) and a stronger notion of (AIP) linked to the no-free lunch condition. We show in a one step model that under (AIP) the super-hedging cost is just the payoff's concave envelop. In the multiple-period case, for a particular, but still general setup, we propose a recursive scheme for the computation of a the super-hedging cost of a convex option .
After revision:  We propose several characterizations of the (AIP ) condition and also study the relation with the classical notions of no-arbitrage .