Embedding metric spaces into normed spaces and estimates of metric capacity

Let M-d be an arbitrary real normed space of finite dimension d >= 2. We define the metric capacity of M-d as the maximal m is an element of N such that every m-point metric space is isometric to some subset of M-d (with metric induced by M-d). We obtain that the metric capacity of M-d lies in the range from 3 to [3/2 d]+1, where the lower bound is sharp for all d; and the upper bound is shown to be sharp for d is an element of {2,3}. Thus, the unknown sharp upper bound is asymptotically linear, since it lies in the range from d+2 to [3/2 d]+ 1.