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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.

60D05 Geometric probability and stochastic geometry
Full Text: DOI
[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
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