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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 -dimensional set by the sizes of its -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

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Fix a dimension and consider the projections

For each 1 ≤ jd, let

Then the Loomis–Whitney inequality holds:

Equivalently, taking we have

implying

A special case

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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).[1]

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 in the th 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[1]

Proof

Overview: We prove it for unions of unit cubes on the integer grid, then take the continuum limit. When , it is obvious. Now induct on . The only trick is to use Hölder's inequality for counting measures.

Enumerate the dimensions of as .

Given unit cubes on the integer grid in , with their union being , we project them to the 0-th coordinate. Each unit cube projects to an integer unit interval on . Now define the following:

  • enumerate all such integer unit intervals on the 0-th coordinate.
  • Let be the set of all unit cubes that projects to .
  • Let be the area of , with .
  • Let be the volume of . We have , and .
  • Let be for all .
  • Let be the area of . We have .

By induction on each slice of , we have

Multiplying by , we have

Thus

Now, the sum-product can be written as an integral over counting measure, allowing us to perform Holder's inequality:

Plugging in , we get

Corollary. Since , we get a loose isoperimetric inequality:

Iterating the theorem yields and more generally[2]where enumerates over all projections of to its dimensional subspaces.

Generalizations

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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

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  1. ^ a b Loomis, L. H.; Whitney, H. (1949). "An inequality related to the isoperimetric inequality". Bulletin of the American Mathematical Society. 55 (10): 961–962. doi:10.1090/S0002-9904-1949-09320-5. ISSN 0273-0979.
  2. ^ Burago, Yurii D.; Zalgaller, Viktor A. (2013-03-14). Geometric Inequalities. Springer Science & Business Media. p. 95. ISBN 978-3-662-07441-1.

Sources

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