In this talk I will discuss how to apply the S-matrix bootstrap maximization program to the 2d bosonic O(N) integrable model which has N species of scalar particles with mass m and no bound states. We show that the known S-matrix follows from maximizing a linear functional in the space of S-matrices allowed by the analyticity, unitarity, and crossing constraints and without using any input from integrability. In fact, we argue that such space of allowed S-matrices is a convex space with vertices. The O(N) model lies at one of those vertices, other vertices corresponding to the free theory and at least one other integrable theory. Therefore the O(N) model (and any other theory at a vertex) maximizes an infinite number of linear functionals such that their gradient points in the general direction of the vertex. This “vertex property" is the reason why the S-matrix of such theories can be computed without any other input than the basic properties of an S-matrix.  Based on arXiv:1805.02812 with Yifei He and Andy Irrgang.