  In this paper we characterize the extremal points of the unit ball of the
Benamou--Brenier energy and of a coercive generalization of it, both subjected
to the homogeneous continuity equation constraint. We prove that extremal
points consist of pairs of measures concentrated on absolutely continuous
curves which are characteristics of the continuity equation. Then, we apply
this result to provide a representation formula for sparse solutions of dynamic
inverse problems with finite dimensional data and optimal-transport based
regularization.
