Functional Loomis-Whitney inequality (${\int }_{{\mathbb{R}}^d}{\prod }_{j=1}^d{f}_j({\hat{x}}_j{)}^{1/(d-1)}\le {\prod }_{j=1}^d\Vert f_j{\Vert }_1^{1/(d-1)}$)

Each nonnegative integrable $f_j$ depends on all coordinates except the $j$th. In dimension three, Cauchy-Schwarz inequality first in the third coordinate and then in the remaining pair proves the displayed estimate. Induction integrates the last coordinate by Hölder's inequality on the first $d-1$ factors, then applies Hölder's inequality to the remaining factor and the dimension-$(d-1)$ product. Taking indicators of coordinate projections of a Euclidean body gives the volume version of the Loomis--Whitney inequality.