Hypermetric Spaces and the Hamming Cone

where / ^ 0, F is a proper subset of {1, 2, . . . , n) and the symbol 1_ is used for "exclusive or": i JL j £ V means i G V, j £ V or i (? V,j(z V. The metrics (2) are extreme rays of the metric cone and are called Hamming rays. The convex hull of these rays is called the Hamming cone Hn and we call d Hamming, if d Ç Hn. Such metrics are also called L-embeddable (e.g., [2]) or addressable (e.g., [5]). Let Q be a finite set and let {A t\l ^ i ^ n) be a collection of n subsets of 12 that will be called addresses. Then it can be shown that a metric d is Hamming if and only if for some finite set 12, there exist addresses A t and non-negative weights Wj (j G 12) so that