For a closed d-dimensional subvariety X of an abelian variety A and a
canonically metrized line bundle L on A, Chambert-Loir has introduced measures
$c_1(L|_X)^{\wedge d}$ on the Berkovich analytic space associated to A with
respect to the discrete valuation of the ground field. In this paper, we give
an explicit description of these canonical measures in terms of convex
geometry. We use a generalization of the tropicalization related to the Raynaud
extension of A and Mumford's construction. The results have applications to the
equidistribution of small points.