Generalized triangle inequalities in \(\mathbb{R}^n\) (Q2765163)

For a list \(L=a_1,\dots, a_m,b_1,\dots, b_p\) of \(m+p\) (not necessarily distinct) points in \(\mathbb{R}^n\) satisfying \(|m-p|\leq k\), a \(k\)-simplex \(\Delta\) is the convex hull of \(k+1\) points taken from \(L\), where also simplices with \(k\)-volume \(|\Delta |_k=0\) are taken into consideration. Thus, writing \(S\) for such a \(k\)-subset of \(L\), the corresponding simplex is denoted by \(\Delta(S)\), and it is said to be of type \((i;k+1-i)\) if precisely \(i\) elements of \(S\) are from \(\{a_1, \dots,a_m\}\) (and the remaining ones from \(\{b_1, \dots,b_m\})\).NEWLINENEWLINENEWLINECombining discrete and continuous methods, the authors generalize the triangle inequality by proving that NEWLINE\[NEWLINE\sum_{ \Delta (S)\text{ of type }(k;1)}\bigl|\Delta(S)\bigr |_k=\sum_{\Delta (S) \text{ of type }(k+1;0)} \bigl|\Delta(S)\bigr |_k.NEWLINE\]NEWLINE Based on this, they also present a proof of the well-known fact that \(\mathbb{R}^n\) is a hypermetric space.