The subindependence of coordinate slabs inl p n balls

Abstract

It is proved that if the probabilityP is normalised Lebesgue measure on one of thelpn balls in Rn, then for any sequencet1, t2, …, tnof positive numbers, the coordinate slabs {|xi|≤ti} 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 thel1n ball which is inside the cube[−1, t]n is less than or equal tofn(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.