Loomis–Whitney inequality explained

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 Lynn Harold Loomis and Hassler Whitney, and was published in 1949.

Statement of the inequality

Fix a dimension

d\ge 2 and consider the projections\pi _{j} : R ^{d} \to R ^{d} , \pi _{j} : x = (x _{1} , ... , x _{d} ) \mapsto \hat{x} _{j} = (x _{1} , ... , x _{j} , x _{j} , ... , x _{d} ).

For each 1 ≤ j ≤ d, let

g _{j} : R ^{d} \to [ 0 , + inf ty), g _{j} \in L ^{d} ( R ^{d} ).

Then the Loomis–Whitney inequality holds:

\left\|\prod ^{d} _{j =1} g _{j} \circ \pi _{j\right\|} ^{L ^{1} ( R ^{d} )} = \int _{R ^{d}} \prod ^{d} _{j =1} g _{j} ( \pi _{j} (x) ) d x \leq \prod ^{d} _{j =1} \| g _{j} \| _{L ^{d} ( R ^{d} )} .

Equivalently, taking

f _{j} (x) = g _{j} (x) ^{d} , we havef _{j} : R ^{d} \to [ 0 , + inf ty), f _{j} \in L ^{1} ( R ^{d} )

implying

\int _{R ^{d}} \prod ^{d} _{j =1} f _{j} ( \pi _{j} (x) ) ^{1} d x \leq \prod ^{d} _{j =1} \left( \int _{R ^{d}} f _{j} (\hat{x} _{j} ) d \hat{x} _{j} \right) ^{1} .

A special case

R ^{d} 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

R ^{d} and letf _{j} = 1 _{\pi _{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,

\prod ^{d} _{j =1} f _{j} (\pi _{j} (x)) ^{1} = \prod ^{d} _{j =1} 1 = 1 .

Hence, by the Loomis–Whitney inequality,

\int _{R ^{d}} 1 _{E(x)} d x = | E | \leq \prod ^{d} _{j =1} | \pi _{j} (E) | ^{1} ,

and hence

| E | \geq \prod ^{d} _{j =1} (| E |)/(| \pi _{j} (E) |) .

The quantity

(| E |)/(| \pi _{j} (E) |)

can be thought of as the average width of

E in the j 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

Corollary. Since

2 |\pi _{j(E)|} \leq |\partial E| , we get a loose isoperimetric inequality:

$$|E|^\backslash leq\; 2^|\backslash partial\; E|^d$$Iterating the theorem yields

| E | \leq \prod _{1} | \pi _{j} \circ \pi _{k} (E) | ^{\binom{d-1} { 2 } ^{-1} } and more generally^[2] $$|\; E\; |\; \backslash leq\; \backslash prod\_j\; |\; \backslash pi\_\; (E)\; |^$$where \pi _{j} enumerates over all projections of \R ^{d} to its d - k dimensional subspaces.

Generalizations

The Loomis–Whitney inequality is a special case of the Brascamp–Lieb inequality, in which the projections π_j above are replaced by more general linear maps, not necessarily all mapping onto spaces of the same dimension.

Sources

• Book: 3524748. Alon. Noga. Spencer. Joel H.. The probabilistic method. Fourth edition of 1992 original. Wiley Series in Discrete Mathematics and Optimization. John Wiley & Sons, Inc.. Hoboken, NJ. 2016. 978-1-119-06195-3. Noga Alon. Joel H. Spencer. 1333.05001.
• Book: 3185193. Boucheron. Stéphane. Lugosi. Gábor. Massart. Pascal. Pascal Massart. Concentration inequalities. A nonasymptotic theory of independence. Oxford University Press. Oxford. 2013. 978-0-19-953525-5. 1279.60005. 10.1093/acprof:oso/9780199535255.001.0001.
• Book: 0936419. Burago. Yu. D.. Zalgaller. V. A.. Geometric inequalities. A. B.. Sosinskiĭ. Grundlehren der mathematischen Wissenschaften. 285. Springer-Verlag. Berlin. 1988. 3-540-13615-0. Yuri Burago. Victor Zalgaller. 10.1007/978-3-662-07441-1. 0633.53002.
• Book: 0102775. Hadwiger. H.. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag. Berlin–Göttingen–Heidelberg. 1957. 0078.35703. 3-642-94702-6. Grundlehren der mathematischen Wissenschaften. 93. 10.1007/978-3-642-94702-5. Hugo Hadwiger.
• Loomis. L. H.. Lynn Harold Loomis. Whitney. H.. Hassler Whitney . An inequality related to the isoperimetric inequality. Bulletin of the American Mathematical Society. 55. 10. 1949. 961–962. 10.1090/S0002-9904-1949-09320-5. free. 0031538. 0035.38302.

Notes and References

1. Loomis . L. H. . Whitney . H. . 1949 . An inequality related to the isoperimetric inequality . Bulletin of the American Mathematical Society . en . 55 . 10 . 961–962 . 10.1090/S0002-9904-1949-09320-5 . 0273-0979. free .
2. Book: Burago . Yurii D. . Geometric Inequalities . Zalgaller . Viktor A. . 2013-03-14 . Springer Science & Business Media . 978-3-662-07441-1 . 95 . en.