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Let \(d \in \mathbb{R}^{E_{n}}\) and \((V_{n}, d)\) be the associated distance space. TFAE:

  1. \(d = \sum_{S \subseteq V_{n}} \lambda_{S} \delta(S)\) for some nonnegative integers \(\lambda_{S}.\)
  2. \((V_{n}, d)\) is hypercube embeddable, \emph{i.e.} there exist \(n\) vectors \(u_{1}, \ldots, u_{n} \in \{ 0,1 \}^m\) (for some \(m\)) such that \(d_{ij} = \left\| u_{i} - u_{j} \right\|_{1}\) for all \(1 \leq i < j \leq n.\)