Averaging distances in finite dimensional normed spaces and John's ellipsoid期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Aicke Hinrichs

Institution:	Mathematisches Institut, FSU Jena, D 07743 Jena, Germany

Abstract:	A Banach space has the average distance property (ADP) if there exists a unique real number such that for each positive integer and all $x_1,\ldots,x_n$ in the unit sphere of there is some in the unit sphere of such that $\begin{displaymath}\frac{1}{n}\sum_{k=1}^n\Vert x_k-x\Vert=r.\end{displaymath}$ A theorem of Gross implies that every finite dimensional normed space has the average distance property. We show that, if has dimension , 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 .

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