Loomis–Whitney inequality

(Redirected from 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 -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

edit

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

edit

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

edit

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

edit
  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

edit