The authors 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, the authors study the Arakelov geometry of toric varieties. In particular, they consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. They show that these notions can be translated in terms of convex analysis and are closely related to objects such as polyhedral complexes, concave functions, real MongeAmpére measures, and LegendreFenchel duality. The authors also present a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This formula allows them to compute the height of toric varieties with respect to some interesting metrics arising from polytopes and compute the height of toric projective curves with respect to the FubiniStudy metric and the height of some toric bundles. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in toric varieties. Reviews "... [T]he exposition is systematically clear, including details on the geometry of toric varieties, metrized line bundles, and metrics and measures on toric varieties."  MAA Reviews Table of Contents  Introduction
 Conventions and notations
 Metrized line bundles and their associated heights
 The LegendreFenchel duality
 Toric varieties
 Metrics and measures on toric varieties
 Height of toric varieties
 Metrics from polytopes
 Variations on FubiniStudy metrics
 Bibliography
 List of Symbols
 Index
