Finite metrics in switching classes (Q858301)

Let \(D\) be a finite set, \(\sigma: D\to{\mathbb R}\) any function, and \(g:D\times D\to {\mathbb R}\) a function satisfying the conditions \(g(x,x)=0\) and \(g(x,y)=g(y,x)\). The authors introduce the function \(g^\sigma\) by the equalities \(g^\sigma(x,x)=0\) and \(g^\sigma(x,y)=\sigma(x)+g(x,y)+\sigma(y)\) if \(x\neq y\). The authors fix \(g\) and consider the set of all possible \(g^\sigma\), denoted by \([g]\). The paper is devoted to the structure of subsets of \([g]\) consisting of metrics, semimetrics, and \(L^1\)-embeddable semimetrics (cut semimetrics).