The setup is as in [this question](https://mathoverflow.net/questions/89762/bounding-the-commutator-a-b-in-terms-of-the-numerical-radius?rq=1):  

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](https://mathoverflow.net/q/90571)).


if $N$ is the [induced $p$-norm](https://en.wikipedia.org/wiki/Matrix_norm#Induced_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 for *all* $p\ge1$?**
> 
> 
> **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.)



Bounding the matrix norm of a commutator $[A,B]$ in terms of the norms of $A$ and $B$
A somewhat more general setting, namely, finding the best constant $C\_{p,q,r}$ in
\begin{equation\*}
 \|AB-BA\|\_p \le C\_{p,q,r}\|A\|\_q\|B\|\_r,
\end{equation\*}
for Schatten $p$,$q$,$r$-norms, is studied in [this paper.](http://www.sciencedirect.com/science/article/pii/S0024379510003356)


*EDIT (1st Mar'17).* See also [this recent paper](http://www.sciencedirect.com/science/article/pii/S0024379517300605) (LAA, **521**(15), May 2017, Pages 263–282) that provides sharp bounds on the constant $C\_{p,q,r}$ above for real matrices.


