In non-archimedean analysis, there is no analogue known for the first Chern form of a metrized line bundle. However, 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. The analogy to the corresponding forms in differential geometry comes from Arakelov geometry. The main result of this paper is an explicit description of these canonical measures \(c_{1}(L|_{X})^{\wedge d}\) in terms of convex geometry. The author uses 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.