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 {displaystyle d} -dimensional set by the sizes of its ( d − 1 ) {displaystyle (d-1)} -dimensional projections. The inequality has applications in incidence geometry, the study of so-called 'lattice animals', and other areas. 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 {displaystyle d} -dimensional set by the sizes of its ( d − 1 ) {displaystyle (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. Fix a dimension d ≥ 2 {displaystyle dgeq 2} and consider the projections For each 1 ≤ j ≤ d, let Then the Loomis–Whitney inequality holds: