Examples for permutation invariant norms I am looking for nice (concrete) examples of permutation invariant norms on $\mathbb{R}^n.$ It is clear that the $\ell_p$ norms do the job. Could you mention me other ones?
 A: Take any bounded convex set $C \subset \mathbb R^n$ whose interior contains $0$. Let $g$ be its Minkowski gauge: $g(x) = \inf \{ t > 0: x \in t C\}$.
Then you get a symmetric norm on $\mathbb R^n$ by 
$$\|x\| = \max \{ g(s T x) \; : \; T \in P_n,\; s \in \{1,-1\}\}$$
where $P_n$ is the set of $n \times n$ permutation matrices.
Moreover, this gives you all the permutation invariant norms.
Well, maybe that's not concrete enough for you.  Here's a way to construct lots of
 examples.
Given any $m$ symmetric norms 
$\|\cdot \|_1, \ldots, \|\cdot \|_j$ on $\mathbb R^n$ and 
a norm $\|\cdot \|_R $ on $\mathbb R^m$ such that $$ \|(x_1, \ldots, x_m)\|_R \le \|(y_1, \ldots, y_m)\|_R \ \text{whenever all}\ 0 \le x_i \le y_i $$
define
$$ \|x \| = \|(\|x\|_1, \ldots, \|x\|_m)\|_R$$ 
So for example, for any $p_0, \ldots, p_m \ge 1$ you could take
$$ \|x \| = \left( \sum_{i=1}^m \|x \|_{p_i}^{p_0} \right)^{1/p_0}
= \left( \sum_{i=1}^m \left( \sum_{j=1}^n |x_j|^{p_i}\right)^{p_0/p_i}\right)^{1/p_0} $$
