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Loomis–Whitney inequality

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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.

Contents

The result is named after the American mathematicians L. H. Loomis and Hassler Whitney, and was published in 1949.

Statement of the inequality

Fix a dimension d ≥ 2 and consider the projections

π j : R d → R d − 1 , π j : x = ( x 1 , … , x d ) ↦ x ^ j = ( x 1 , … , x j − 1 , x j + 1 , … , x d ) .

For each 1 ≤ j ≤ d, let

g j : R d − 1 → [ 0 , + ∞ ) , g j ∈ L d − 1 ( R d − 1 ) .

Then the Loomis–Whitney inequality holds:

∫ R d ∏ j = 1 d g j ( π j ( x ) ) d x ≤ ∏ j = 1 d ∥ g j ∥ L d − 1 ( R d − 1 ) .

Equivalently, taking

f j ( x ) = g j ( x ) d − 1 , ∫ R d ∏ j = 1 d f j ( π j ( x ) ) 1 / ( d − 1 ) d x ≤ ∏ j = 1 d ( ∫ R d − 1 f j ( x ^ j ) d x ^ j ) 1 / ( d − 1 ) .

A special case

The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space R d to its "average widths" in the coordinate directions. Let E be some measurable subset of R d and let

f j = 1 π j ( E )

be the indicator function of the projection of E onto the jth coordinate hyperplane. It follows that for any point x in E,

∏ j = 1 d f j ( π j ( x ) ) 1 / ( d − 1 ) = 1.

Hence, by the Loomis–Whitney inequality,

| E | ≤ ∏ j = 1 d | π j ( E ) | 1 / ( d − 1 ) ,

and hence

| E | ≥ ∏ j = 1 d | E | | π j ( E ) | .

The quantity

| E | | π j ( E ) |

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.

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.

References

Loomis–Whitney inequality Wikipedia
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