  We study the geodesic problem on the group of diffeomorphism of a domain
M$\subset$Rd, equipped with the H(div) metric. The geodesic equations coincide
with the Camassa-Holm equation when d=1, and represent one of its possible
multi-dimensional generalizations when d>1. We propose a relaxation {\`a} la
Brenier of this problem, in which solutions are represented as probability
measures on the space of continuous paths on the cone over M. We use this
relaxation to prove that smooth H(div) geodesics are globally length minimizing
for short times. We also prove that there exists a unique pressure field
associated to solutions of our relaxation. Finally, we propose a numerical
scheme to construct generalized solutions on the cone and present some
numerical results illustrating the relation between the generalized
Camassa-Holm and incompressible Euler solutions.
