Is there a generalization of jung's theorem to $l_p$ norms? Let $S\subseteq \mathbb{R}^d$ for $d>1$ be compact and define 
$$ diam(S,p)=\max_{x,y\in S} \|x-y\|_p $$
where for $x\in\mathbb{R}^d$  the $l_p$-norm is defined as
$$\|x\|_p:=\left(\sum_{i=1}^d x_i^p\right)^{\tfrac{1}{p}}.$$
Furthermore, let $B_p(r):=\{ x\in \mathbb{R}^d \colon \|x\|_p=r\}$ and let $r_p^*(S)$ be the minimal $r$ such that $S$ can be enclosed by a translate of $B_p(r)$. 
Then Jung's Theorem states that for $p=2$ we have
$$ r_2^*(S) \leq \left(\tfrac{d}{2(d+1)}\right)^{\tfrac{1}{2}}  diam(S,2).$$
I would like to know if there is a more general formula for all $p\in[1,\infty)$ that reduces to this formula for $p=2$, for example, something like
$$ r_p^*(S) \leq \left(\tfrac{d}{2(d+1)}\right)^{\tfrac{1}{p}}  diam(S,p).$$

I've found a result stating
$$ r^*_p(S) \leq \tfrac{d}{d+1} diam(S,p) $$
which holds true even if you replace the $l_p$ norms by arbitrary gauge functions in a minkowski geometry in
Leichtweiss, Kurt. "Zwei Extremalprobleme der Minkowski-Geometrie.." Mathematische Zeitschrift 62 (1955): 37-49
Since there are gauge functions for which this is sharp, that'll have to do for now, even if I'd appreciate a closer bound.
