In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a d-dimensional set by the sizes of its (d-1)-dimensional projections. The inequality has applications in incidence geometry, the study of so-called "lattice animals", and other areas. The result is named after the American mathematicians Lynn Harold Loomis and Hassler Whitney, and was published in 1949.

Statement of the inequality

Fix a dimension d\ge 2 and consider the projections For each 1 ≤ j ≤ d, let Then the Loomis–Whitney inequality holds: Equivalently, taking we have implying

A special case

The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space to its "average widths" in the coordinate directions. This is in fact the original version published by Loomis and Whitney in 1949 (the above is a generalization). Let E be some measurable subset of and let be the indicator function of the projection of E onto the jth coordinate hyperplane. It follows that for any point x in E, Hence, by the Loomis–Whitney inequality, and hence The quantity can be thought of as the average width of E in the jth coordinate direction. This interpretation of the Loomis–Whitney inequality also holds if we consider a finite subset of Euclidean space and replace Lebesgue measure by counting measure. The following proof is the original one Corollary. Since, we get a loose isoperimetric inequality: Iterating the theorem yields and more generally where \pi_j enumerates over all projections of \R^d to its d-k dimensional subspaces.

Generalizations

The Loomis–Whitney inequality is a special case of the Brascamp–Lieb inequality, in which the projections πj above are replaced by more general linear maps, not necessarily all mapping onto spaces of the same dimension.

Sources