A generalization of the concept of distance based on the simplex inequality (Q1646650)

For a nonempty set \(X\) and for an \(n\)-distance on \(X\), that is, a function \(d: X^n \to { \mathbb{R}}_{+}\) satisfying {\parindent=0.8cm \begin{itemize}\item[(i)] \(d(x_1, \dots, x_n ) \leq \sum_{j=1}^n d(x_1, \dots, x_n )^z_j\) for all \(x_1, \dots, x_n, z\in X\), \item[(ii)] \(d(x_1, \dots, x_n) = d(x_{\sigma(1)}, \dots, x_{\sigma(n)})\) for all \(x_1, \dots, x_n \in X\) and all \(\sigma \in S_n\), \item[(iii)] \(d(x_1, \dots, x_n) = 0\) if and only if \(x_1, \dots, x_n\), \end{itemize}} the authors consider the set of real numbers \(K \in (0, 1]\) for which the condition \[ d(x_1, \dots, x_n) \leq K \sum_{j=1}^n d(x_1, \dots, x_n )^z_j , \qquad x_1, \dots, x_n, z\in X\tag{1} \] holds. Here \(d(x_1, \dots, x_n)^z_j\) is obtained from \(d(x_1, \dots, x_n)\) by setting its \(j\)-th variable to \(z\) and \(S_n\) is the symmetry group. The authors provide natural examples of \(n\)-distances and investigate the infimum \(K^*\) of \(K\) for which the inequality (1) holds for each of them, and introduce a generalization of the concept of \(n\)-distance by replacing the sum function in \((i)\) with an arbitrary symmetric \(n\)-variable function \(g : {\mathbb{R}^n_{+}} \to {\mathbb{R}}_{+}\). The paper is fundamental and useful to study metric spaces and inequalities.