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 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 L. H. Loomis and Hassler Whitney, and was published in 1949.

Statement of the inequality

Fix a dimension d ≥ 2 and consider the projections

\pi _{{j}}:{\mathbb {R}}^{{d}}\to {\mathbb {R}}^{{d-1}},

\pi _{{j}}:x=(x_{{1}},\dots ,x_{{d}})\mapsto {\hat {x}}_{{j}}=(x_{{1}},\dots ,x_{{j-1}},x_{{j+1}},\dots ,x_{{d}}).

For each 1 ≤ j ≤ d, let

g_{{j}}:{\mathbb {R}}^{{d-1}}\to [0,+\infty ),

g_{{j}}\in L^{{d-1}}({\mathbb {R}}^{{d-1}}).

Then the Loomis–Whitney inequality holds:

\int _{{{\mathbb {R}}^{{d}}}}\prod _{{j=1}}^{{d}}g_{{j}}(\pi _{{j}}(x))\,{\mathrm {d}}x\leq \prod _{{j=1}}^{{d}}\|g_{{j}}\|_{{L^{{d-1}}({\mathbb {R}}^{{d-1}})}}.

Equivalently, taking

f_{{j}}(x)=g_{{j}}(x)^{{d-1}},

\int _{{{\mathbb {R}}^{{d}}}}\prod _{{j=1}}^{{d}}f_{{j}}(\pi _{{j}}(x))^{{1/(d-1)}}\,{\mathrm {d}}x\leq \prod _{{j=1}}^{{d}}\left(\int _{{{\mathbb {R}}^{{d-1}}}}f_{{j}}({\hat {x}}_{{j}})\,{\mathrm {d}}{\hat {x}}_{{j}}\right)^{{1/(d-1)}}.

A special case

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

f_{{j}}={\mathbf {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 _{{j=1}}^{{d}}f_{{j}}(\pi _{{j}}(x))^{{1/(d-1)}}=1.

Hence, by the Loomis–Whitney inequality,

|E|\leq \prod _{{j=1}}^{{d}}|\pi _{{j}}(E)|^{{1/(d-1)}},

and hence

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

The quantity

{\frac {|E|}{|\pi _{{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 π_j above are replaced by more general linear maps, not necessarily all mapping onto spaces of the same dimension.

References

Loomis, Lynn H.; Whitney, H. (1949). "An inequality related to the isoperimetric inequality". Bull. Amer. Math. Soc. 55: 961–962. doi:10.1090/S0002-9904-1949-09320-5. MR 0031538

This article is issued from Wikipedia - version of the 6/18/2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.