  We consider the scattering matrices of massive quantum field theories with no
bound states and a global $O(N)$ symmetry in two spacetime dimensions. In
particular we explore the space of two-to-two S-matrices of particles of mass
$m$ transforming in the vector representation as restricted by the general
conditions of unitarity, crossing, analyticity and $O(N)$ symmetry. We found a
rich structure in that space by using convex maximization and in particular its
convex dual minimization problem. At the boundary of the allowed space special
geometric points such as vertices were found to correspond to integrable
models. The dual convex minimization problem provides a novel and useful
approach to the problem allowing, for example, to prove that generically the
S-matrices so obtained saturate unitarity and, in some cases, that they are at
vertices of the allowed space.
