$L^p$ norm means Consider the unit sphere $S_p^{n-1}$ of an $L^p$ normin $\mathbb{R}^n.$ The question is: what is the expected value of the  $L^q$ norm on $S_p^{n-1}?$ Since (I assume) this is intractable in closed form, what are the asymptotics in $q, n$ (for $p$ fixed)?
 A: For $p=2$: up to multiplicative universal constants, the average $M$ of  $\|\cdot\|_q$ over $S^{n-1}$ is equal to 


*

*$M \simeq \sqrt{q} \cdot n^{1/q-1/2}$ when $1 \leq q \leq \log n$,

*$M \simeq \sqrt{\log n}/\sqrt{n}$ for $q \geq \log n$.


This can be checked most easily after switching to a Gaussian integral as Yemon mentions. For the lower bound in 1, it may be useful to consider using concentration of measure. If it is a matter of reference, it can probably be extracted from Chapter 5.4 in Milman-Schechtman, "Asymptotic theory of finite-dimensional normed spaces". Indeed, the value of this average is closely related to the dimension of almost Euclidean sections of the space $\ell_q^n$.
Edit: let me add more detail. First, (this is true for any norm of $\mathbb{R}^n$, just by rotational invariance of the Gaussian measure $\gamma_n$), we have
$$ M = \frac{1}{\alpha_n} \int_{\mathbb{R}^n} \|x\|_q \, \mathrm{d} \gamma_n(x), $$
where 
$$\alpha_n = \int_{\mathbb{R}^n} \|x\|_2 \, \mathrm{d} \gamma_n(x) $$
is a constant very close to $\sqrt{n}$. Now write
$$ M \leq \frac{1}{\alpha_n} \left(\int_{\mathbb{R}^n} \|x\|^q_q \, \mathrm{d} \gamma_n(x) \right)^{1/q} \simeq  \sqrt{q} \cdot n^{1/q-1/2} $$
(use the fact the $L^q$ norm of a standard Gaussian variable is or order $\sqrt{q}$). This upper bound is sharp when $q \leq \log n$, this follows from concentration of measure. Finally for $q \geq \log n$, the norms $\|\cdot\|_q$ and $\|\cdot\|_{\infty}$ are equivalent, and the question reduces to estimating the expected maximum of $n$ i.i.d. standard Gaussian variables.
A: Igor, pending finding the original or the version in W+W, here are my notes, let me stick to $\mathbb R^3;$ evidently there were also a bunch of lower case Greek letters that i set to 1 here: Given $x,y,z > 0$ and
$$  x^p + y^q + z^r < 1, $$ we get
$$  \int x^{a-1} y^{b-1} z^{c-1} dx dy dz $$ as
$$   \frac{ \Gamma\left( \frac{a}{p} \right) \Gamma\left( \frac{b}{q} \right) \Gamma\left( \frac{c}{r} \right)   }{\Gamma\left( 1 + \frac{a}{p} + \frac{b}{q}+ \frac{c}{r}\right) }  $$
Dirichlet, Über eine neue Methode zur Bestimmung vielfacher Integrale, original 1839 
A: I don't know if you are very interested in this, but one special case where we can work out the average explicitly is $q=1$. In this case,
$$
\frac{1}{S_{n-1}}\int_{S_{n-1}}\|x\|_1\, d\sigma(x) = \frac{n}{S_{n-1}}\int_{S_{n-1}} |x_1|\, d\sigma(x) = \frac{2n S_{n-2}}{S_{n-1}} \int_0^1 x (1-x^2)^{(n-3)/2}\, dx\\
=\frac{2nS_{n-2}}{(n-1)S_{n-1}} = \frac{2n}{n-1}\, \frac{\Gamma(\frac{n-1}{2})}{\pi^{1/2}\Gamma(\frac{n}{2}-1)}.
$$
Here, I write $S_d$ for both the $d$-dimensional unit sphere and its surface area. (It so happens I did the same calculation earlier on this site, in a different context.)
If $n=2k+2$, say, then this equals
$$
\frac{4k+4}{2k+1}\, \frac{k(2k)!}{4^k(k!)^2} \sim \frac{2}{\pi^{1/2}}\, k^{1/2} \sim \sqrt{\frac{2}{\pi}}\, n^{1/2},
$$
by Stirling's formula to obtain the asymptotics.
(This asymptotic behavior is of course consistent with the intuition suggested by Yemon in the comments.)
