It is a known fact that for any $2\neq p\in[1,\infty]$, the linear isometries for the corresponding norm $\|\cdot\|_p$ on $\mathbb{R}^d$ is the set of all square-matrices with entries in $\{-1,1,0\}$, with at exactly one nonzero entry for each line and each column, that is : the hypercube group.

Question: what are the norms that precisely have this set of matrices as linear isometry group ?

When trying to exhibit such norms $N$, the most general construction that I could guess is the following : \begin{equation} N(x) = \int_1^{+\infty} \|x\|_p \,\mathrm{d}\mu, \end{equation} where $\mu$ is a finite measure over $[1,+\infty)$. I expect that this should cover all possible such norms, but was not able to prove it.

One track would be to use some Choquet theory for which the previous formula would describe $N$ as a generalized linear combination of extreme points (the norms $\|\cdot\|_p$), but I am not much acquainted with such point of view.

I welcome any reference or suggestion on this question !

Thanks,

Ayman