On the best constants associated with \(n\)-distances (Q2204109)

Let \(X\) be an arbitrary set, with \(\vert X\vert \ge 2\), let \(n\ge 2\) be an integer and \(\mathbb{R}_+=[0,+\infty]\). A map \(d:X^n \to \mathbb{R}_+\) is called \(n\)-distance on \(X\) if it satisfies the following conditions:\par (i) \(d(x_1,\cdots,x_n)=0\) if \(x_1=\cdots =x_n\);\par (ii) \(d\) is invariant under any permutation of its arguments;\par (iii) \(d(x_1,\cdots,x_n)\le \sum_{i=1}^n d(x_1,\cdots,x_n)_i^z\) for all \(x_1,\cdots,x_n,z \in X\), \par where \(d(x_1,\cdots,x_n)_i^z\) stands for the function obtained from \(d(x_1,\cdots,x_n)\) by setting the \(i\)th variable to \(z\).\par Condition (iii) is called simplex inequality. For many \(n\)-distances the simplex inequality can be refined into \[d(x_1,\cdots,x_n)\le K_n \sum_{i=1}^n d(x_1,\cdots,x_n)_i^z, \quad x_1,\cdots,x_n,z \in X,\] for some constant \(K_n \in [0,1]\), the minimum possible \(K_n^*\) is called optimal (or best) constant.\par In Section 2 of the paper, several examples of \(n\)-distances are given with their optimal constants when possible, or an interval enclosing the best constant. \par Section 3 deals with the so-called partial simplex inequality, that is an inequality of the form \[ d(x_1,\cdots,x_n)\le K_{n,k}\sum_{i=1}^k d(x_1,\cdots,x_n)_i^z, \quad x_1,\cdots,x_n,z \in X,\] for some \(k\in \{2,\cdots,n-1\}\) and some \(K_{n,k}>0\); as before \(K_{n,k}^*\) denote the optimal constant. Some relations between \(K_n^*\) and \(K_{n,k}^*\) are proved for specific \(n\)-distances.\par Section 4 presents additional properties for certain subclasses of \(n\)-distances.\par The notion of multidistance is presented in Section 5: a multidistance on \(X\) is a function \(d:\bigcup_{n\ge 2} X^n \to \mathbb{R}_+\) such that\par (i) \(d(x_1,\cdots,x_n)=0\) if \(x_1=\cdots =x_n\);\par (ii) \(d_{\vert X^n}\) is invariant under any permutation of its arguments;\par (iii) \(d(x_1,\cdots,x_n)\le \sum_{i=1}^n d(x_i,z)\) for all \(x_1,\cdots,x_n,z \in X\).\par Relations between multidistances and \(n\)-distances are proved.