On the volume of the intersection of two $L_p^n$ balls

Gideon Schechtman; Joel Zinn

paper

On the volume of the intersection of two $L_{p}^{n}$ balls

arXiv:math/9201206

Abstract

This note deals with the following problem, the case $p = 1$ , $q = 2$ of which was introduced to us by Vitali Milman: What is the volume left in the $L_{p}^{n}$ ball after removing a t-multiple of the $L_{q}^{n}$ ball? Recall that the $L_{r}^{n}$ ball is the set ${(t_{1}, t_{2}, \dots, t_{n}); t_{i} \in R, n^{- 1} \sum_{i = 1}^{n} ∣ t_{i} ∣^{r} \leq 1}$ and note that for $0 < p < q < \infty$ the $L_{q}^{n}$ ball is contained in the $L_{p}^{n}$ ball. In Corollary 4 we show that, after normalizing Lebesgue measure so that the volume of the $L_{p}^{n}$ ball is one, the answer to the problem above is of order $e^{- c t^{p} n^{p / q}}$ for $T < t < \frac{1}{2} n^{\frac{1}{p} - \frac{1}{q}}$ , where $c$ and $T$ depend on $p$ and $q$ but not on $n$ . The main theorem, Theorem 3, deals with the corresponding question for the surface measure of the $L_{p}^{n}$ sphere.

On the volume of the intersection of two Lpn​ balls

Abstract

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On the volume of the intersection of two $L_{p}^{n}$ balls