[Paper Review] On the volume of the intersection of two $L_p^n$ balls

Gideon Schechtman, Joel Zinn|ArXiv.org|Nov 9, 1989

Mathematics and ApplicationsMathematics1 references19 citations

TL;DR

This paper establishes sharp exponential bounds for the volume of the intersection of $L_p^n$ and $L_q^n$ balls in high dimensions, showing that the normalized volume outside a $t$-multiple of the $L_q^n$ ball decays as $\exp(-c t^p n^{p/q})$ for $t$ in a specific range. The analysis uses a novel class of $L_p$-distributed random variables and surface measure techniques to derive precise tail estimates for $\ell_p^n$-norms.

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 $\{(t_1,t_2,\dots,t_n);\ t_i\in{\bf R},\ n^{-1}\sum_{i=1}^n|t_i|^r\le 1\}$ and note that for $0

Motivation & Objective

• To determine the asymptotic volume of the $L_p^n$ ball not contained in a $t$-scaled $L_q^n$ ball for $1 \leq p < q < \infty$.
• To derive precise tail estimates for the $\ell_p^n$-norm under the normalized surface measure on the $L_p^n$ sphere.
• To establish universal constants in the exponential decay rate of the volume outside the $L_q^n$ ball, independent of $n$.
• To extend the results to the normalized Lebesgue measure on the $L_p^n$ ball via integration over radial components.

Proposed method

• Introduces a class of i.i.d. random variables with density $c_p e^{-t^p}$, whose normalized $\ell_p^n$-norm vector is uniformly distributed on the positive quadrant of the $L_p^n$ sphere.
• Uses the independence of the normalized vector from the radial component $S = (\sum x_i^p)^{1/p}$ to decouple angular and radial behavior.
• Applies moment and tail bounds on $x_i^p$ using properties of the gamma function and exponential moments, especially $\mathbb{E}[e^{-h x^p}] = (1+h)^{-1/p}$.
• Employs a chaining argument with dyadic decomposition of indices to control the $\ell_q^n$-norm of the normalized vector, using bounds on $\max x_i$ and $\mathbb{E}[\sum x_i^q]^{1/q}$.
• Derives upper and lower tail bounds for $\|u\|_{L_q^n}$ under surface measure using exponential moment inequalities and concentration estimates.
• Transfers results from surface measure to volume measure via the formula $\nu(A) = n \int_0^1 r^{n-1} \mu(A/r) \, dr$, where $\nu$ is normalized Lebesgue measure on the $L_p^n$ ball.

Experimental results

Research questions

• RQ1 What is the asymptotic volume of the $L_p^n$ ball that lies outside a $t$-scaled $L_q^n$ ball for $1 \leq p < q < \infty$?
• RQ2 How do the tail probabilities of the $\ell_q^n$-norm behave on the $L_p^n$ sphere under normalized surface measure?
• RQ3 Can the decay rate of the volume outside the $L_q^n$ ball be quantified with constants independent of $n$?
• RQ4 What is the precise dependence of the decay rate on $t$, $p$, $q$, and $n$?
• RQ5 How do the results for surface measure extend to the full volume measure on the $L_p^n$ ball?

Key findings

• For $t > T(p,q)$, the normalized surface measure of the set where $\|u\|_{L_q^n} > t$ is bounded above by $\exp(-c t^p n^{p/q})$, with $c$ depending only on $p$ and $q$.
• For $2 \leq t \leq \frac{1}{2} n^{1/p - 1/q}$, the normalized surface measure satisfies $\mu(\|u\|_{L_q^n} > t) \geq \exp(-C t^p n^{p/q})$, with $C$ depending only on $p$ and $q$.
• The constants $c$ and $C$ in the upper and lower bounds can be taken as $\gamma / p$ and $\Gamma / p$ respectively for $q > 2p$, with universal $\gamma, \Gamma$.
• For $q = \infty$, the tail bounds become $\mu(\|u\|_\infty > t) \leq \exp(-\gamma t^p / p)$ and $\geq \exp(-\Gamma t^p / p)$ for $t > \tau (\log n)^{1/p}$.
• The normalized volume of the $L_p^n$ ball outside a $t$-scaled $L_q^n$ ball decays as $\exp(-c t^p n^{p/q})$ for $t > T(p,q)$, with $T(p,q) = \tau \min\{q, \log n\}^{1/p}$.
• The results extend to $p > 0$ with modified thresholds, though the constants' dependence on $p$ and $q$ is not fully quantified in that case.