The setup is as in this question:

Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that
$$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M}_n(\mathbb C).$$

Equivalently, $C_N$ is the maximum of $N(AB-BA)$ provided that $N(A)=N(B)=1$.

Given examples of $C_N$ are

- $C_N=\sqrt{2}$ if $N$ is the Frobenius norm
- $C_N=2$ if $N$ is the operator norm $\| \cdot\|_2$
- $C_N=4$ if $N$ is the
*numerical radius*$r(A)=\sup\limits_{x\ne0}\dfrac{|x^*Ax|}{\|x\|^2}$ (See this answer to an MO question).

if $N$ is the induced $p$-norm, defined for $1\le p\le\infty$ by $\|A\| _p = \sup \limits _{x \ne 0} \frac{\| A x\| _p}{\|x\|_p}$, we have $C_N=2$ for $p=\infty$ (with $\|A\|_\infty $ being just the maximum absolute row sum of the matrix). Indeed, the lower bound $2$ for $\|\cdot\|_\infty $ is obtained by taking e.g. $A=\begin{pmatrix} 1&0\\1&0\end{pmatrix}$ and $B=\begin{pmatrix} 0&1\\0&-1\end{pmatrix}$, and it should be easy to prove that $2$ is also the general upper bound for $\|\cdot\|_\infty $.

Similarly, $C_N=2$ for $p=1$ (with $\|A\|_1 $ being the maximum absolute column sum of the matrix).

Knowing that $C_N\equiv2$ for $p=1,2,\infty$, is it true that the same holds for the induced $p$-norms forall$p\ge1$?

If $N$ runs over all possible matrix norms, what is the range of $C_N$? In particular, is it bounded below and/or above?

(To avoid trivialities, let's keep it homogeneous by only considering "normalized" norms, i.e. require $N(I_n)=1$. This does not seem to be part of the standard definition of a norm.)