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Loomis–Whitney inequality

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 L. H. Loomis and Hassler Whitney, and was published in 1949.

Contents

Statement of the inequalityEdit

Fix a dimension d ≥ 2 and consider the projections

 
 

For each 1 ≤ jd, let

 
 

Then the Loomis–Whitney inequality holds:

 

Equivalently, taking

 
 

A special caseEdit

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. 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.

GeneralizationsEdit

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.

ReferencesEdit