Equality in the three-dimensional Loomis--Whitney inequality forces a measurable body to agree up to a null set with a Cartesian product . This follows from the equality conditions in the two Cauchy-Schwarz inequalities used in its proof. If the body is connected and is a finite union of positive-volume axis-parallel boxes, each is an interval and equality up to a null set upgrades to exact equality with one box.
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The Loomis–Whitney inequality is a geometric inequality in the field of differential geometry and convex analysis. It provides a relationship between the volume of a convex body in Euclidean space and the volumes of its projections onto lower-dimensional spaces.