Finiteness of the Hölder–Brascamp–Lieb constant revisited

Abstract Abstract Hölder–Brascamp–Lieb inequalities have become a ubiquitous tool in Fourier analysis in recent years, due in large part to a theorem of Bennett et al. characterizing finiteness of the Hölder–Brascamp–Lieb constant. Here we provide a new characterization of a substantially different nature involving directed graphs of subspaces. Its practical value derives from its complementary nature to the Bennett et al. conditions: it creates a means by which one can establish finiteness of the Hölder–Brascamp–Lieb constant by analysis of a well‐chosen, finite list of subspaces rather than by checking conditions on all subspaces of the underlying vector space. The proof is elementary and is essentially an “explicitization” of the semi‐explicit factorization algorithm of Carbery et al.