On the volume of the intersection of two $L_p^n$ balls
| dc.creator | Schechtman, Gideon | |
| dc.creator | Zinn, Joel | |
| dc.date | 1989-11-09 | |
| dc.date.accessioned | 2026-07-07T09:14:40Z | |
| dc.date.available | 2026-07-07T09:14:40Z | |
| dc.description | 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 $\{(t_1,t_2,\dots,t_n);\ t_i\in{\bf R},\ n^{-1}\sum_{i=1}^n|t_i|^r\le 1\}$ 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 $e^{-ct^pn^{p/q}}$ for $T<t<{1\over 2}n^ {{1\over p}-{1\over q}}$, 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. | |
| dc.identifier | https://arxiv.org/abs/math/9201206 | |
| dc.identifier | http://arxiv.org/abs/math/9201206 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/152756 | |
| dc.subject | Functional Analysis | |
| dc.subject | Metric Geometry | |
| dc.subject | 52A | |
| dc.title | On the volume of the intersection of two $L_p^n$ balls | |
| dc.type | text |