  It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the
first author, that the euclidean Brascamp-Lieb inequality satisfies a natural
and useful Fourier duality property. The purpose of this paper is to establish
an appropriate discrete analogue of this. Our main result identifies the
Brascamp-Lieb constants on (finitely-generated) discrete abelian groups with
Brascamp-Lieb constants on their (Pontryagin) duals. As will become apparent,
the natural setting for this duality principle is that of locally compact
abelian groups, and this raises basic questions about Brascamp-Lieb constants
formulated in this generality.
