Let $U \subseteq \mathbf{R}^d$ be the unit ball with respect to the $l_p$-norm. I would like find the minimal $m=m(d)$ such that that there is a simplex $S$ with $S\subseteq U \subseteq mS$.

For the $l_2$-norm, $m(d)=d$, i.e it is equal to the dimension. I am not sure how this varies for several other norms. Even an upper bound for $m$ is good. Can we get something like $d^{p-1}$ perhaps?