On the volume of the intersection of two \(L^ n_ p\) balls. (English) Zbl 0704.60017

Schechtman, G.; Zinn, J.

Proc. Am. Math. Soc. 110, No. 1, 217-224 (1990).

Note that for \(0<p<q<\infty\) the \(L^ n_ p\)-ball \[ \{(t_ 1,...,t_ n)\in {\mathbb{R}}^ n:\;(1/n)\sum^{n}_{i=1}| t_ i|^ p\leq 1\} \] contains the \(L^ n_ q\)-ball. The authors prove that the volume left in the \(L^ n_ p\)-ball after removing a t-multiple of the \(L^ n_ q\)-ball is of order \(\exp (-ct^ pn^{p/q})\). For this aim they exploit properties of i.i.d. random variables with density function \((p/\Gamma (1/p))\exp (-t^ p)\), \(t>0\), which are related to \(L_ p\) in the same way as Gaussian variables are related to \(L_ 2\).

Reviewer: M.Zähle

MSC:

60E15

Inequalities; stochastic orderings

Keywords:

volume of the intersection of two L(p,n) balls; distributions; Gaussian variables