Some Thoughts on Infinity Norm « LOUIS’S MATH BLOG

Posted by: louisyangliu | March 5, 2009

Some Thoughts on Infinity Norm

The infinity norm may be one of the usual models to study about in different situations in mathematics, not the continuous ones, but the discrete or singular ones.

We know that $||(x,y)||_{\infty}:=max(|x|,|y|)$ , for any $(x,y)\in \mathbb{R}^{2}$ , can be expressed as an integral on $S^{1}$ by spreading a measure on it, which is

$||(x,y)||_{\infty}=c\int_{S^{1}}|x\cos\theta+y\sin\theta|(\delta_{\frac{\pi}{4}}(\theta)+\delta_{\frac{3\pi}{4}}(\theta))d\theta$ (1)

for some constant $c$ because $max(|x|,|y|)=\frac{1}{2}(|x+y|+|x-y|)$ , where

$\delta_{\frac{\pi}{4}}(\theta)=\begin{cases}<br /> +\infty, & \theta=\frac{\pi}{4}\\<br /> 0, & \theta\ne\frac{\pi}{4}.\end{cases}$ (2)

satisfying $\int_{-\infty}^{\infty}\delta_{\frac{\pi}{4}}(\theta)\, dx=1$ and

$\delta_{\frac{3\pi}{4}}(\theta)=\begin{cases}<br /> +\infty, & \theta=\frac{3\pi}{4}\\<br /> 0, & \theta\ne\frac{3\pi}{4}.\end{cases}$ (3)

satisfying $\int_{-\infty}^{\infty}\delta_{\frac{3\pi}{4}}(\theta)\, dx=1$ are modified Dirac delta functions.

One might also be able to put an measure on $S^{3}$ for the complex norm $||(z,w)||_{\infty}=max(|z|,|w|)$ , $(z,w)\in \mathbb{C}^{2}$ , which can be written in real norm as $||(x,y,u,v)||:=max(\sqrt{x^{2}+y^{2}},\sqrt{u^{2}+v^{2}})$ . A problem is what function $f$ would satisfy

$||(x,y)||_{\infty}=\int_{S^{3}}|x\cos\xi_{1}\cos\eta+y\cos\xi_{2}\sin\eta|f(\xi_{1},\xi_{2},\eta)d\xi_{1}d\xi_{2}d\eta$ (4)

if one parametrizes $S^{3}$ by Hopf coordinates

$z=e^{i\xi_{1}}\sin\eta,\, w=e^{i\xi_{2}}\cos\eta.$ (5)

One can show that the function $f$ is actually some function independent of $\xi_{1}$ and $\xi_{2}$ by the invariance of the complex norm under $U(1)\times U(1)$ action, so it can be denoted as $f(\eta)$ . But it is unknown yet what exactly the function $f(\eta)$ is, because the invariance yields that $f(\eta)=\delta_{\frac{\pi}{4}}(\eta)+\delta_{\frac{3\pi}{4}}(\eta)$ , but this doesn’t produce an appropriate function in the general expression for the complex infinity norm. If one looks back the case of real infinity norm, the way of expressing the infinity norm is by making use of the absolute values on sum and difference smartly to sift out the maximum from two number. But in the complex case the double integral on the torus