In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a -dimensional set by the sizes of its -dimensional projections. The inequality has applications in incidence geometry, the study of so-called "lattice animals", and other areas.
The Loomis–Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space 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]
be the indicator function of the projection of E onto the jth 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 in the 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.
Overview: We prove it for unions of unit cubes on the integer grid, then take the continuum limit. When , it is obvious. Now induct on . The only trick is to use Hölder's inequality for counting measures.
Enumerate the dimensions of as .
Given unit cubes on the integer grid in , with their union being , we project them to the 0-th coordinate. Each unit cube projects to an integer unit interval on . Now define the following:
enumerate all such integer unit intervals on the 0-th coordinate.
Let be the set of all unit cubes that projects to .
Let be the area of , with .
Let be the volume of . We have , and .
Let be for all .
Let be the area of . We have .
By induction on each slice of , we have
Multiplying by , we have
Thus
Now, the sum-product can be written as an integral over counting measure, allowing us to perform Holder's inequality:
Plugging in , we get
Corollary. Since , we get a loose isoperimetric inequality:
Iterating the theorem yields and more generally[2]
where enumerates over all projections of to its dimensional subspaces.
Generalizationsedit
The Loomis–Whitney inequality is a special case of the Brascamp–Lieb inequality, in which the projections πj above are replaced by more general linear maps, not necessarily all mapping onto spaces of the same dimension.
Referencesedit
^ abLoomis, L. H.; Whitney, H. (1949). "An inequality related to the isoperimetric inequality". Bulletin of the American Mathematical Society. 55 (10): 961–962. doi:10.1090/S0002-9904-1949-09320-5. ISSN 0273-0979.
^Burago, Yurii D.; Zalgaller, Viktor A. (2013-03-14). Geometric Inequalities. Springer Science & Business Media. p. 95. ISBN978-3-662-07441-1.
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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 Contents 1 Statement of the inequality 2 A special case 3 Generalizations 4 References 5 SourcesStatement of the inequality editFix a dimension d 2 displaystyle d geq 2 nbsp and consider the projections p j R d R d 1 displaystyle pi j mathbb R d to mathbb R d 1 nbsp p 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 nbsp For each 1 j d let g j R d 1 0 displaystyle g j mathbb R d 1 to 0 infty nbsp g j L d 1 R d 1 displaystyle g j in L d 1 mathbb R d 1 nbsp Then the Loomis Whitney inequality holds j 1 d g j p j L 1 R d R d j 1 d g j p 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 nbsp Equivalently taking f j x g j x d 1 displaystyle f j x g j x d 1 nbsp we have f j R d 1 0 displaystyle f j mathbb R d 1 to 0 infty nbsp f j L 1 R d 1 displaystyle f j in L 1 mathbb R d 1 nbsp implying R d j 1 d f j p j x 1 d 1 d x j 1 d R d 1 f j x j d x j 1 d 1 displaystyle 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 nbsp A special case editThe Loomis Whitney inequality can be used to relate the Lebesgue measure of a subset of Euclidean space R d displaystyle mathbb R d nbsp 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 displaystyle mathbb R d nbsp and let f j 1 p j E displaystyle f j mathbf 1 pi j E nbsp be the indicator function of the projection of E onto the jth coordinate hyperplane It follows that for any point x in E j 1 d f j p j x 1 d 1 1 displaystyle prod j 1 d f j pi j x 1 d 1 1 nbsp Hence by the Loomis Whitney inequality E j 1 d p j E 1 d 1 displaystyle E leq prod j 1 d pi j E 1 d 1 nbsp and hence E j 1 d E p j E displaystyle E geq prod j 1 d frac E pi j E nbsp The quantity E p j E displaystyle frac E pi j E nbsp can be thought of as the average width of E displaystyle E nbsp in the j displaystyle j nbsp 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 1 Proof Overview We prove it for unions of unit cubes on the integer grid then take the continuum limit When d 1 2 displaystyle d 1 2 nbsp it is obvious Now induct on d 1 displaystyle d 1 nbsp The only trick is to use Holder s inequality for counting measures Enumerate the dimensions of R d 1 displaystyle mathbb R d 1 nbsp as 0 1 d displaystyle 0 1 d nbsp Given N displaystyle N nbsp unit cubes on the integer grid in R d 1 displaystyle mathbb R d 1 nbsp with their union being T displaystyle T nbsp we project them to the 0 th coordinate Each unit cube projects to an integer unit interval on R displaystyle mathbb R nbsp Now define the following I 1 I k displaystyle I 1 I k nbsp enumerate all such integer unit intervals on the 0 th coordinate Let T i displaystyle T i nbsp be the set of all unit cubes that projects to I i displaystyle I i nbsp Let N j displaystyle N j nbsp be the area of p j T displaystyle pi j T nbsp with j 0 1 d displaystyle j 0 1 d nbsp Let a i displaystyle a i nbsp be the volume of T i displaystyle T i nbsp We have i a i N displaystyle sum i a i N nbsp and a i N 0 displaystyle a i leq N 0 nbsp Let T i j displaystyle T ij nbsp be p j T i displaystyle pi j T i nbsp for all j 1 d displaystyle j 1 d nbsp Let a i j displaystyle a ij nbsp be the area of T i j displaystyle T ij nbsp We have i a i j N j displaystyle sum i a ij N j nbsp By induction on each slice of T i displaystyle T i nbsp we have a i d 1 j 1 d a i j displaystyle a i d 1 leq prod j 1 d a ij nbsp Multiplying by a i N 0 displaystyle a i leq N 0 nbsp we have a i d N 0 j 1 d a i j displaystyle a i d leq N 0 prod j 1 d a ij nbsp ThusN i a i i N 0 1 d j 1 d a i j 1 d N 0 1 d i 1 k j 1 d a i j 1 d displaystyle N sum i a i leq sum i N 0 1 d prod j 1 d a ij 1 d N 0 1 d sum i 1 k prod j 1 d a ij 1 d nbsp Now the sum product can be written as an integral over counting measure allowing us to perform Holder s inequality i 1 k j 1 d a i j 1 d i j 1 d a i j 1 d j 1 d a j 1 d 1 j a j 1 d d j 1 d i 1 k a i j 1 d displaystyle sum i 1 k prod j 1 d a ij 1 d int i prod j 1 d a ij 1 d left prod j 1 d a cdot j 1 d right 1 leq prod j a cdot j 1 d d prod j 1 d left sum i 1 k a ij right 1 d nbsp Plugging in i a i j N j displaystyle sum i a ij N j nbsp we get N d j 0 d N j displaystyle N d leq prod j 0 d N j nbsp Corollary Since 2 p j E E displaystyle 2 pi j E leq partial E nbsp we get a loose isoperimetric inequality E d 1 2 d E d displaystyle E d 1 leq 2 d partial E d nbsp Iterating the theorem yields E 1 j lt k d p j p k E d 1 2 1 displaystyle E leq prod 1 leq j lt k leq d pi j circ pi k E binom d 1 2 1 nbsp and more generally 2 E j p j E d 1 k 1 displaystyle E leq prod j pi j E binom d 1 k 1 nbsp where p j displaystyle pi j nbsp enumerates over all projections of R d displaystyle mathbb R d nbsp to its d k displaystyle d k nbsp dimensional subspaces Generalizations editThe Loomis Whitney inequality is a special case of the Brascamp Lieb inequality in which the projections pj above are replaced by more general linear maps not necessarily all mapping onto spaces of the same dimension References edit a b Loomis L H Whitney H 1949 An inequality related to the isoperimetric inequality Bulletin of the American Mathematical Society 55 10 961 962 doi 10 1090 S0002 9904 1949 09320 5 ISSN 0273 0979 Burago Yurii D Zalgaller Viktor A 2013 03 14 Geometric Inequalities Springer Science amp Business Media p 95 ISBN 978 3 662 07441 1 Sources editAlon Noga Spencer Joel H 2016 The probabilistic method Wiley Series in Discrete Mathematics and Optimization Fourth edition of 1992 original ed Hoboken NJ John Wiley amp Sons Inc ISBN 978 1 119 06195 3 MR 3524748 Zbl 1333 05001 Boucheron Stephane Lugosi Gabor Massart Pascal 2013 Concentration inequalities A nonasymptotic theory of independence Oxford Oxford University Press doi 10 1093 acprof oso 9780199535255 001 0001 ISBN 978 0 19 953525 5 MR 3185193 Zbl 1279 60005 Burago Yu D Zalgaller V A 1988 Geometric inequalities Grundlehren der mathematischen Wissenschaften Vol 285 Translated by Sosinskiĭ A B Berlin Springer Verlag doi 10 1007 978 3 662 07441 1 ISBN 3 540 13615 0 MR 0936419 Zbl 0633 53002 Hadwiger H 1957 Vorlesungen uber Inhalt Oberflache und Isoperimetrie Grundlehren der mathematischen Wissenschaften Vol 93 Berlin Gottingen Heidelberg Springer Verlag doi 10 1007 978 3 642 94702 5 ISBN 3 642 94702 6 MR 0102775 Zbl 0078 35703 Loomis L H Whitney H 1949 An inequality related to the isoperimetric inequality Bulletin of the American Mathematical Society 55 10 961 962 doi 10 1090 S0002 9904 1949 09320 5 MR 0031538 Zbl 0035 38302 Retrieved from https en wikipedia org w index php title Loomis Whitney inequality amp oldid 1184174976, wikipedia, wiki, book, books, library,