Embedding Metric Spaces into Normed Spaces and Estimates of Metric Capacity

Let M be an arbitrary real normed space of finite dimension d ≥ 2. We define the metric capacity of M as the maximal m ∈ N such that every m-point metric space is isometric to some subset of M (with metric induced by M). We obtain that the metric capacity of M lies in the range from 3 to ⌊ 3 2d ⌋ + 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 ⌊ 3 2d ⌋ + 1. Mathematics Subject Classification (AMS 2000): 52A21, 52B10, 52C10