Averaging distances in real quasihypermetric Banach spaces of finite dimension (Q1293978)

The surface of the unit ball \(S(E)\) in a finite dimensional normed space \(E\) is an interesting metric space. The author [Arch. Math. 62, No. 4, 338-344 (1994; Zbl 0821.46018)] has defined the ``rendezvous number'' \(r(E)\) of \(E\) to be that unique number [whose existence is ensured by a remarkable theorem of \textit{O. Gross} in Ann. Math. Stud. 52, 49-53 (1964; Zbl 0126.16401); see also \textit{J. Cleary, S. A. Morris} and \textit{D. Yost}, Am. Math. Mon. 93, 260-275 (1986; Zbl 0598.51014) for a beautiful survey] such that for each finite subset \(\{x_i: i=1,2,\ldots ,k\}\) of \(S(E)\) there exists \(y\) in \(S(E)\) with \(kr(E) = \sum \|x_i - y \|\). The conjecture is that if \(\dim E = n\) then \(r(E) \leq 2 - n^{-1}\) (with equality known for \(\ell_1(n)\)). Here the author proves the conjecture for the case when \(E\) is ``quasi-hypermetric'', \textit{i.e.} the unit ball in \(E\) is the dual of a zonoid or, equivalently, \(E\) is isometric to a subspace of \(L_1[0,1].\) Note that the metric on \(S(E)\) is the norm metric. A more natural metric for \(S(E)\) is the intrinsic metric induced by the norm [see, for example, \textit{J. J. Schäffer}, ``Geometry of spheres in normed spaces'', Lect. Notes Pure Appl. Math. 20, Marcel Dekker (1976; Zbl 0344.46038)] and one wonders about the values of the rendezvous number in this metric.