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 $n$ for the largest cardinality of a set $S$ of unit vectors in an $n$-dimensional Minkowski space (or Banach space) such that the sum of any subset has norm less than 1? We prove that $|S|\leq 2n$ and that equality holds iff the space is linearly isometric to $\ell ^{n}_{\infty }$, the space with an $n$-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.