  We present an explicit expression for the normalized height of a projective
toric variety. This expression decomposes as a sum of local contributions, each
term being the integral of a certain function, concave and piecewise
linear-affine. More generally, we obtain an explicit expression for the
normalized multiheight of a torus with respect to several monomial embeddings.
  The set of functions introduced behaves as an arithmetic analog of the
polytope classically associated with the torus action. Besides the formulae for
the height and multiheight, we show that this object behaves in a natural way
with respect to several standard constructions: decomposition into orbits,
joins, Segre products and Veronese embeddings.
  The proof follows an indirect way: instead of the definition of the
normalized height, we rely on the computation of an appropriate arithmetic
Hilbert function.
