Averaging distances in finite dimensional normed spaces and John's ellipsoid

A Banach space $X$ has the average distance property (ADP) if there exists a unique real number $r=r(X)$ such that for each positive integer $n$ and all $x_1,\ldots,x_n$ in the unit sphere of $X$ there is some $x$ in the unit sphere of $X$ such that

A theorem of Gross implies that every finite dimensional normed space has the average distance property. We show that, if $X$ has dimension $d$, then $r(X) \le 2-1/d$. This is optimal and answers a question of Wolf (Arch. Math., 1994). The proof is based on properties of the John ellipsoid of maximal volume contained in the unit ball of $X$.