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 formula_1-dimensional set by the sizes of its formula_2-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 formula_3 and consider the projections For each 1 ≤ "j" ≤ "d", let Then the Loomis–Whitney inequality holds : Equivalently, taking A special case. The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space formula_11 to its "average widths" in the coordinate directions. Let "E" be some measurable subset of formula_11 and let be the indicator function of the projection of "E" onto the "j"th 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 formula_18 in the formula_19th 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" above are replaced by more general linear maps, not necessarily all mapping onto spaces of the same dimension.