On affine invariant and local Loomis–Whitney type inequalities

Abstract

We prove various extensions of the Loomis-Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors w i of a not necessarily orthonormal basis of R n , or their orthogonal complements. In order to prove such inequalities we estimate the constant in the Brascamp-Lieb inequality in terms of the vectors w i. Restricted and functional versions of the inequality will also be considered.