Posted by: Louis Y. Liu | March 1, 2011

On the Infinity Norm

The infinity norm is often used as a model in analysis and geometry, not for continuous scenarios, but for the discrete or singular scenarios.

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

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

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

$latex \delta_{\frac{3\pi}{4}}(\theta)=\begin{cases}
+\infty, & \theta=\frac{3\pi}{4}\\
0, & \theta\ne\frac{3\pi}{4}.\end{cases}$ (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 a 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 the 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

\left\{ (\frac{\sqrt{2}}{2}e^{i\xi_{1}},\frac{\sqrt{2}}{2}e^{i\xi_{2}}):\xi_{1},\xi_{2}\in[0,2\pi]\right\} (6)

eliminates the feasibility of sifting out the maximal modulus.

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