TITLE: "Scaled $L^p$ norm" and geometric mean
QUESTION [36 upvotes]: The $L^p$ norm in $\mathbb{R}^n$ is
\begin{align}
\|x\|_p = \left(\sum_{j=1}^{n} |x_j|^p\right)^{1/p}.
\end{align}
Playing around with WolframAlpha, I noticed that, if we define the "scaled" $L^p$ norm in $\mathbb{R}^n$ to be
\begin{align}
\| x \|_p = \left(\frac{1}{n}\sum_{j=1}^{n} |x_j|^p\right)^{1/p}
\end{align}
then
\begin{align}
\lim_{p \to 0} \|x\|_p &= \left( \prod_{j=1}^{n} |x_j| \right)^{1/n},
\end{align}
which is the geometric mean of the coordinates' absolute values. This is interesting maybe because the $L^p$ norm doesn't have a nice limit at zero.
My questions:

How do I prove this?
Is this definition of "scaled $L^p$ norm" interesting, or known by another name, or used anywhere?
Is there any interesting reason to define the $L^0$ norm as the geometric mean, as above?
Further reading?

Thanks!

REPLY [11 votes]: Scaled $L^p$ norm (or rather $\ell^p$, since you work with vectors) is known as Generalized mean. A bunch of interesting inequalities involving the means are found in the book Inequalities by Hardy, Littlewood, and Pólya. 

The integral geometric mean $\exp\left(\frac{1}{2\pi}\int_0^{2\pi} \log f(\theta)\,d\theta \right)$ comes up in complex analysis, especially as it relates to operator theory and involves the name of Gabor Szegő. See the terse Wikipedia article on Szegő limit theorems and the not-at-all-terse book by Barry Simon Szego's Theorem and Its Descendants

In a visit back to his native Budapest, Pólya mentioned this conjecture to Szegő, then an undergraduate, and he proved the theorem below, published in 1915... At the time, Szegő was nineteen, and when the paper was published, he was serving in the Austrian Army in World War I

The book Banach spaces of analytic functions by Kenneth Hoffman presents this topic from the viewpoint of complex analysis without much operator theory. The quantity $\exp\left(\frac{1}{2\pi}\int_0^{2\pi} \log f(\theta)\,d\theta \right)$ turns out to be equal to $\inf_{p}\int|1-p|^2 f(\theta)\,d\theta$ where $p$ runs over all polynomials vanishing at $0$. In particular, this gives a criterion for the density of polynomials in weighted $L^2$ spaces.

In a rather different direction, the integral geometric mean comes up in number theory. If $p$ a complex polynomial, the quantity $\exp\left(\frac{1}{2\pi}\int_0^{2\pi} \log |p(\theta)|\,d\theta \right)$ is called the Mahler measure of $p$, denoted $M(p)$. Lehmer's conjecture asserts that there is a gap $(1,\mu)$ in the possible values of $M(p)$: that is, either $M(p)=1$ or $M(p)\ge \mu>1$. Conjecturally, $\mu$ is attained by the polynomial $$p(z)= z^{10}-z^9+z^7-z^6+z^5-z^4+z^3-z+1$$ But even the existence of such $\mu$ remains unknown, let alone its precise value. The Wikipedia article has a good list of references.

That said, please do not call the integral geometric mean "the $L^0$ norm". This term is ambiguous and misleading enough as it is.