Embedding metric spaces into normed spaces and estimates of metric capacity

Abstract

Let \({\cal M}^d\) be an arbitrary real normed space of finite dimension d ≥ 2. We define the metric capacity of \({\cal M}^d\) as the maximal \(m \in {\Bbb N}\) such that every m-point metric space is isometric to some subset of \({\cal M}^d\) (with metric induced by \({\cal M}^d\)). We obtain that the metric capacity of \({\cal M}^d\) lies in the range from 3 to \(\left\lfloor\frac{3}{2}d\right\rfloor+1\), where the lower bound is sharp for all d, and the upper bound is shown to be sharp for d ∈ {2, 3}. Thus, the unknown sharp upper bound is asymptotically linear, since it lies in the range from d + 2 to \(\left\lfloor\frac{3}{2}d\right\rfloor+1\).