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.
Statement of the inequality Fix a dimension d ≥ 2 {\displaystyle d\geq 2} and consider the projections
π j : R d → R d − 1 , {\displaystyle \pi _{j}:\mathbb {R} ^{d}\to \mathbb {R} ^{d-1},}
π j : x = ( x 1 , … , x d ) ↦ x ^ j = ( x 1 , … , x j − 1 , x j + 1 , … , x d ) . {\displaystyle \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 : R d − 1 → [ 0 , + ∞ ) , {\displaystyle g_{j}:\mathbb {R} ^{d-1}\to [0,+\infty ),}
g j ∈ L d − 1 ( R d − 1 ) . {\displaystyle g_{j}\in L^{d-1}(\mathbb {R} ^{d-1}).}
Then the Loomis–Whitney inequality holds:
‖ ∏ j = 1 d g j ∘ π j ‖ L 1 ( R d ) = ∫ R d ∏ j = 1 d g j ( π j ( x ) ) d x ≤ ∏ j = 1 d ‖ g j ‖ L d − 1 ( R d − 1 ) . {\displaystyle \left\|\prod _{j=1}^{d}g_{j}\circ \pi _{j}\right\|_{L^{1}(\mathbb {R} ^{d})}=\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 , {\displaystyle f_{j}(x)=g_{j}(x)^{d-1},} we have
f j : R d − 1 → [ 0 , + ∞ ) , {\displaystyle f_{j}:\mathbb {R} ^{d-1}\to [0,+\infty ),}
f j ∈ L 1 ( R d − 1 ) {\displaystyle f_{j}\in L^{1}(\mathbb {R} ^{d-1})}
implying
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