Averaging distances in finite dimensional normed spaces and John's ellipsoid (Q2758997)

A Banach space \(X\) has average distance property if there exists a unique real number \(r=r(X)>0\) (a rendezvous number) such that for each \(n\in {\mathbb N}\) there is an \(x\in S\) such that \(\sum^n_{k=1} \|x_k-x\|/n=r\). A theorem of \textit{O.~Gross} [Ann. Math. Stud. 52, 49-53 (1964; Zbl 0126.16401)] states that any compact connected metric space has a unique rendezvous number. It follows that \(r(X)\) exists for every finite-dimensional Banach space \(X\). The main result of the paper proves \textit{R. Wolf}'s conjecture [Arch. Math. 62, No. 4, 338-344 (1994; Zbl 0821.46018)]: \(r(X)\leq 2-1/d\) for every finite-dimensional Banach space \(X\), \(\dim X=d\), the inequality being sharp for \(\ell^1(d)\). The proof is based on properties of the John's ellipsoid of maximal volume contained in the unit ball of~\(X\).