A complete toric variety of dimension is determined by a lattice and a complete integral fan in . This variety has a model over the integers and is equipped with the action of a torus . An equivariant ample line bundle on determines an integral polytope in the dual space . Plenty of algebro-geometric properties of the pair can easily be read off from the polytope . The exponential map determines a parametrization of the open orbit by . Assume that is equipped with a positive Hermitian metric that is equivariant under the action of the compact torus. Then, minus the logarithm of the norm of a section of , determines a strictly convex function on . The stability set of this function turns out to be the polytope and the Legendre dual is a strictly convex function on . This function is the symplectic potential in the Guillemin-Abreu theory. We prove that the height of with respect to the metrized line bundle is given by times the integral of with respect to the normalized Haar measure of . This is the arithmetic analogue of the expression of the degree of a toric variety as times the volume of the polytope. We expect that many other Arakelov geometric properties of can be read from the function g. This is a report on joint work with J.I. Burgos (Barcelone) and P. Philippon (Paris).