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.
Fix a dimension
and consider the projections For each 1 ≤ j ≤ d, let Then the Loomis–Whitney inequality holds: Equivalently, taking
we have implying 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
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] Overview: We prove it for unions of unit cubes on the integer grid, then take the continuum limit.
, it is obvious.
The only trick is to use Hölder's inequality for counting measures.
Enumerate the dimensions of
unit cubes on the integer grid in
, we project them to the 0-th coordinate.
Each unit cube projects to an integer unit interval on
Now define the following: By induction on each slice of
Now, the sum-product can be written as an integral over counting measure, allowing us to perform Holder's inequality:
Corollary.
, we get a loose isoperimetric inequality:
Iterating the theorem yields
enumerates over all projections of
dimensional subspaces.
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.