  H\"older-Brascamp-Lieb inequalities provide upper bounds for a class of
multilinear expressions, in terms of $L^p$ norms of the functions involved.
They have been extensively studied for functions defined on Euclidean spaces.
Bennett-Carbery-Christ-Tao have initiated the study of these inequalities for
discrete Abelian groups and, in terms of suitable data, have characterized the
set of all tuples of exponents for which such an inequality holds for specified
data, as the convex polyhedron defined by a particular finite set of affine
inequalities.
  In this paper we advance the theory of such inequalities for torsion-free
discrete Abelian groups in three respects. The optimal constant in any such
inequality is shown to equal $1$ whenever it is finite. An algorithm that
computes the admissible polyhedron of exponents is developed. It is shown that
nonetheless, existence of an algorithm that computes the full list of
inequalities in the Bennett-Carbery-Christ-Tao description of the admissible
polyhedron for all data, is equivalent to an affirmative solution of Hilbert's
Tenth Problem over the rationals. That problem remains open.
  Applications to computer science will be explored in a forthcoming companion
paper.
