The subindependence of coordinate slabs inl p n balls

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, . 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. Supported in part by NSF DMS-9257020. Supported by a grant from Public Benefit Foundation Alexander S. Onassis. This work will form part of a Ph.D. thesis written by the second-named author.