×

zbMATH — the first resource for mathematics

The subindependence of coordinate slabs in \(l_p^n\) balls. (English) Zbl 0918.60013
Summary: It is proved that if the probability \(P\) is normalised Lebesgue measure on one of the \(l^n_p\) balls in \(\mathbb{R}^n\), then for any sequence \(t_1, t_2, \dots,t_n\) of positive numbers, the coordinate slabs \(\{| x_i| \leq t_i\}\) are subindependent, namely, \[ P\left(\bigcap^n_1\bigl\{| x_i|\leq t_i\bigr\}\right)\leq\prod^n_1P\biggl(\bigl\{| x_i|\leq t_i\bigr\} \biggr). \] A consequence of this result is that the proportion of the volume of the \(l^n_1\) ball which is inside the cube \([-t,t]^n\) is less than or equal to \(f_n(t)= (1-(1-t)^n)^n\). It turns out that this estimate is remarkably accurate over most of the range of values of \(t\). A reverse inequality, demonstrating this, is the second major result of the article.

MSC:
60D05 Geometric probability and stochastic geometry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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. · Zbl 0704.60017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.