If we consider the S-matrix in a subspace of states (for example 2-particle states) and impose the general constraints of unitarity, crossing, analyticity and global symmetries, we find that the allowed space of S-matrices is convex. At the boundary of such convex space we find interesting points such as vertices where in some cases integrable models are located. Such vertices can be found by maximizing linear functionals.
In this talk I will consider the 2d O(N) integrable model and show that it indeed appears at a vertex of the space of allowed S-matrices, discuss how to visualize such space and introduce the so called dual minimization problem that allows to prove several interesting properties of this approach. In particular we show that we always obtain an S-matrix that saturates unitarity (no particle creation). It is an open problem to see if this method leads to interesting results for the world-sheet S-matrix (although the O(6) model is already of interest for AdS/CFT).