= Functional Loomis-Whitney inequality
{title2=$\int_{\mathbb R^d}\prod_{j=1}^df_j(\widehat x_j)^{1/(d-1)}\leq\prod_{j=1}^d\|f_j\|_1^{1/(d-1)}$}
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>.
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