Extremal problems in Minkowski space related to minimal networks

We solve the following problem of Z. Füredi, J. C. Lagarias and F. Morgan (1991): Is there an upper-bound polynomial in for the largest cardinality of a set of unit vectors in an -dimensional Minkowski space (or Banach space) such that the sum of any subset has norm less than 1? We prove that and that equality holds iff the space is linearly isometric to , the space with an -cube as unit ball. We also remark on similar questions they raised that arose out of the study of singularities in length-minimizing networks in Minkowski spaces.