The subindependence of coordinate slabs inl _{p} ^{n} balls
 Keith Ball,
 Irini Perissinaki
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It is proved that if the probabilityP is normalised Lebesgue measure on one of thel _{p} ^{n} balls in R^{ n }, then for any sequencet _{ 1 } , t _{ 2 } , …, t _{ n }of positive numbers, the coordinate slabs {x _{ i } ≤t _{ i }} are subindependent, namely, \(P\left( {\mathop \cap \limits_1^n \{ \left {x_i } \right \leqslant t_i \} } \right) \leqslant \prod\limits_1^n {P(\{ \left {x_i } \right \leqslant t_i \} )} \) . A consequence of this result is that the proportion of the volume of thel _{1} ^{n} ball which is inside the cube[−1, t] ^{ n } is less than or equal tof _{ n } (t)=(1−(1−t) ^{ n } ) ^{ n }. It turns out that this estimate is remarkably accurate over most of the range of values oft. A reverse inequality, demonstrating this, is the second major result of the article.
 G. Schechtman and J. Zinn,On the volume of the intersection of two L _{n} ^{p} balls, Proceedings of the American Mathematical Society110 (1990), 217–224. CrossRef
 Title
 The subindependence of coordinate slabs inl _{p} ^{n} balls
 Journal

Israel Journal of Mathematics
Volume 107, Issue 1 , pp 289299
 Cover Date
 19981201
 DOI
 10.1007/BF02764013
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
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 Authors

 Keith Ball ^{(1)}
 Irini Perissinaki ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University College London, Gower Street, WC1E 6BT, London, U.K.