Norms on $\mathbb{R}^d$ whose linear isometries are the hypercube group 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
 A: Here is a complete answer in dimension 2.
Among all the norms on $\mathbb{R}^2$ which are invariant under the dihedral group $D_8$, the extremal rays are norms whose unit ball are octagons with the required symmetries ($\ell_1$ and $\ell_{\infty}$ appear as octagons which degenerate into squares).
The reason is specific to dimension 2: any norm on $\mathbb{R}^2$ is of form
$$ \|x\| = \int_{S^1} | \langle x , \theta \rangle | d \mu(\theta) $$
for a positive finite measure $\mu$ on $S^1$. (This measure is unique if we require that it is even.) In other words, any 2-dimensional real space embeds into $L^1$. This fails for the $\ell_{\infty}$ norm on $\mathbb{R}^3$. 
Under the required symmetries, the measure $\mu$ is uniquely defined by its restriction to the west-northwest part of the unit circle. Among those, extreme rays are Dirac masses, leading to octagons.
In higher dimensions you get examples of such norms using Orlicz norms. I don't know what are the extreme rays there.
Edit : here are some references, A Class of Convex Bodies
Ethan D. Bolker Transactions of the American Mathematical Society
Vol. 145 (Nov., 1969), pp. 323-345. Schneider, Rolf, and Wolfgang Weil. "Zonoids and related topics." Convexity and its Applications. Birkhäuser Basel, 1983. 296-317.
A: This is not answer, rather a summarize of some remarks.
Let $G$ be the hypercube group. A first step towards classifying all norms with isometry group $G$ is to classify all norms with isometry group which contains $G$. This is what we discuss below.
Let $D$ be the set of all seminorms on $\mathbb{R}^d$ for which the average value on the standard basis vector is 1 (this is a natural normalization). Observe that $D$ is a convex set, compact wrt the pointwise convergence topology.  Observe that $G$ acts naturally on $D$ and let $P:D\to D$ be the associated averaging operator $N(\cdot)\mapsto \frac{1}{|G|}\Sigma_g N(g^{-1}\cdot)$.
Denoting by $C$ the set of $G$ invariant seminorms (in fact: norms, as $G$ acts irreducibly on $\mathbb{R}^d$) in $D$, $P$ is a projection on $C$.
The $P$-preimage of an extreme point in $C$ is a face in $D$, hence contains an extreme point of $D$.
By this observation, it appears that our task is reduced to classifying the set $E$ consisting of all extreme points in $D$.
Given a non-zero linear functional $\theta\in (\mathbb{R}^d)^*$, up to a suitable normalization we have $|\theta(\cdot)|\in D$ and it is easy to see that this is an extreme point. Let us denote by $F\subset E$ the space of extreme points coming from functionals as above.
Guillaume Aubrun made in his answer the following observation:
Observation: A norm in $N\in D$ is in the closed convex hull of $F$ iff $(\mathbb{R}^d,N)$ is embedable in $L^1$.
Proof: 
There exists a probability measure $\mu\in \text{Prob}(F)$ s.t for all $x\in \mathbb{R}^d$, $N(x)=\int |\theta(x)|d\mu(\theta)$
$\Leftrightarrow$ There exists a linear map $\mathbb{R}^d\to L^1(\mu)$, $x\mapsto \theta(x)$.
It then follows that for $d=2$ indeed $E=F$, as every norm on $\mathbb{R}^2$ is embedable in $L^1$ (right? reference?).
However, for $d>2$ there are norms which are not embedable in $L^1$, e.g $\|\cdot\|_\infty$ (also $\|\cdot\|_p$, $p>2$ - correct?), thus $F\subsetneq E$.
What are the other extreme points of $D$?
