Take the tour ×  Let $E^n$ be the real or complex space of
dimension $n$. If $N$ and $M$ are two norms over $E^n$, and if
$A$ is an endomorphism, then
$$\|A\|^M_N:=\sup_{x\ne0}\frac{M(Ax)}{N(x)}$$ is an operator
norm of $A$.  The Banach-Mazur distance between $N$ and $M$ is
$$d(N,M):=\log\inf_{A\in Aut(E)}\|A\|^M_N\|A^{-1}\|^N_M.$$ It is
actually a semi-distance, with $d(N,M)=0$ iff one passes from
$N$ to $M$ by a change of coordinates. Thus $d$ is a distance on
some quotient space, in which there is only one Euclidian norm,
for instance.  Let us define the usual $\ell^p$-norm by
$$\|x\|_p:=\left(\sum_j|x_j|^p\right)^{1/p}.$$ The
$L^\infty$-norm is as usual. It has been known for a long time
that if $1\le p,q\le2$, or if $2\le p,q\le\infty$, then
$$d(\ell^p,\ell^q)=\left|\frac{1}{p}-\frac{1}{q}\right|\log n.$$
What is known if $1\le p\le2\le q\le+\infty$ ?  share|improve
this question add comment  1 Answer  I've just looked in the
"Handbook of the Geometry of Banach Spaces", Chapter 1, Section
8, which gives