  A recurring obstacle in the study of Wasserstein gradient flow is the lack of
convexity of the square Wasserstein metric. In this paper, we develop a class
of transport metrics that have better convexity properties and use these
metrics to prove an Euler-Lagrange equation characterizing Wasserstein discrete
gradient flow. We then apply these results to give a new proof of the
exponential formula for the Wasserstein metric, mirroring Crandall and
Liggett's proof of the corresponding Banach space result. We conclude by using
our approach to give simple proofs of properties of the gradient flow,
including the contracting semigroup property and energy dissipation inequality.
