  We study a one dimensional Lagrangian problem including the variational
reformulation, derived in a recent work of Ambrosio-Baradat-Brenier, of the
discrete Monge-Amp\`ere gravitational model, which describes the motion of
interacting particles whose dynamics is ruled by the optimal transport problem.
The more general action type functional we consider contains a discontinuous
potential term related to the descending slope of the opposite squared distance
function from a generic discrete set in R^{d}. We exploit the underlying
geometrical structure provided by the associated Voronoi decomposition of the
space to obtain C^{1,1} regularity for local minimizers out of a finite number
of shock times.
