Israel Journal of Mathematics

, Volume 107, Issue 1, pp 289–299

The subindependence of coordinate slabs inlpn balls

  • Keith Ball
  • Irini Perissinaki
Article

DOI: 10.1007/BF02764013

Cite this article as:
Ball, K. & Perissinaki, I. Isr. J. Math. (1998) 107: 289. doi:10.1007/BF02764013

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.

Copyright information

© The Magnes Press 1998

Authors and Affiliations

  • Keith Ball
    • 1
  • Irini Perissinaki
    • 1
  1. 1.Department of MathematicsUniversity College LondonLondonU.K.