Hypermetric Spaces and the Hamming Cone

Abstract

We denote by d = (d12, …, d1n, d23, …, dn-1,n) a vector of distances between n points. Such a vector d is called a metric if it satisfies the triangle inequalities(1)The set of all metrics on n points forms a convex polyhedral cone, the extremal properties of which are discussed in [4]. We will be concerned with a sub-cone that is spanned by metrics of the form(2)where t ≧ 0, V is a proper subset of {1, 2, …, n}; and the symbol ⊥ is used for “exclusive or”: i ⊥ j ∈ V means i ∈ V, j ∉ V or i ∉ V,j∉ V.