We show that the height of a toric variety with respect to a toric metrized
line bundle can be expressed as the integral over a polytope of a certain
adelic family of concave functions. To state and prove this result, we study
the Arakelov geometry of toric varieties. In particular, we consider models
over a discrete valuation ring, metrized line bundles, and their associated
measures and heights. We show that these notions can be translated in terms of
convex analysis, and are closely related to objects like polyhedral complexes,
concave functions, real Monge-Amp\`ere measures, and Legendre-Fenchel duality.
We also present a closed formula for the integral over a polytope of a function
of one variable composed with a linear form. This allows us to compute the
height of toric varieties with respect to some interesting metrics arising from
polytopes. We also compute the height of toric projective curves with respect
to the Fubini-Study metric, and of some toric bundles.Comment: Revised version, 230 pages, 3 figure
