All the facets of the six-point Hamming cone (Q1262536)

For any finite set \(X_ n=\{x_ 1,...,x_ n\}\) and for some non-empty \(S\subset \{1,...,n\}\) the symmetric nonnegative function d on \(X_ n\times X_ n\) with \(d(x_ i,x_ k)=t\) for \(i\in S\) and \(k\not\in S\) or \(i\not\in S\) and \(k\in S\) is called Hamming semimetric for each \(t\geq 0\). The convex hull of all points \((d(x_ 1,x_ 2),...,d(x_{n-1},x_ n))\in R^{\left( \begin{matrix} n\\ 2\end{matrix} \right)}\) with Hamming semimetric d is called the Hamming cone \(H_ n\). The authors describe the facets of \(H_ 6\) by so called \((2k+1)\)-gonal inequalities which are generalized triangle inequalities. The results allow the following corollary: For arbitrary six-point semimetric spaces \((X_ n,d)\) the \(L^ 1\)-embeddability is equivalent to: d is 7-gonal, and to: d is hypermetric.