The movement of data (communication) between levels of a memory hierarchy, or between parallel processors on a network, can greatly dominate the cost of computation, so algorithms that minimize communication are of interest. Motivated by this, attainable lower bounds for the amount of communication required by algorithms were established by several groups for a variety of algorithms, including matrix computations. Prior work of Ballard-Demmel-Holtz-Schwartz relied on a geometric inequality of Loomis and Whitney for this purpose. In this paper the general theory of discrete multilinear Holder-Brascamp-Lieb (HBL) inequalities is used to establish communication lower bounds for a much wider class of algorithms. In some cases, algorithms are presented which attain these lower bounds. Several contributions are made to the theory of HBL inequalities proper. The optimal constant in such an inequality for torsion-free Abelian groups is shown to equal one whenever it is finite. Bennett-Carbery-Christ-Tao had characterized the tuples of exponents for which such an inequality is valid as the convex polyhedron defined by a certain finite list of inequalities. The problem of constructing an algorithm to decide whether a given inequality is on this list, is shown to be equivalent to Hilbert's Tenth Problem over the rationals, which remains open. Nonetheless, an algorithm which computes the polyhedron itself is constructed.