  We provide a unifying interpretation of various optimal transport problems as
a minimisation of a linear functional over the set of all Choquet
representations of a given pair of probability measures ordered with respect to
a certain convex cone of functions. This allows us to provide novel proofs of
duality formulae. Among our tools is Strassen's theorem.
  We provide new formulations of the primal and the dual problem for martingale
optimal transport employing a novel representation of the set of extreme points
of probability measures in convex order on Euclidean space. We exhibit a link
to uniformly convex and uniformly smooth functions and provide a new
characterisation of such functions.
  We introduce a notion of martingale triangle inequality. We show that
Kantorovich--Rubinstein duality bears an analogy in the martingale setting
employing the cost functions that satisfy the inequality.
