Inequalities for vertex distances of two simplices (Q703647)

The authors prove several inequalities concerning the vertex distances, the facets areas and the volumes of two simplices. These inequalities are a new contribution to the field of research developed by Pedoe on geometric inequalities for two \(n\)-simplices. In particular they prove that if \(\Omega=\langle A_0,A_1,\dots,A_n\rangle\) and \(\Omega '=\langle A_0',A_1',\dots,A_n'\rangle\) are two \(n\)-simplices with volume \(V\) and \(V'\) and with facet areas \(F_i\) and \(F_i'\) \((i=0,1,\dots,n)\), respectively, \(F=\sum_{i=0}^nF_i\) and \(F'=\sum_{i=0}^nF_i'\), then \[ \begin{aligned} \sum_{i=0}^n \sum_{j=0}^n F_j'(F-2F_i)\| A_iA_j'\| ^2 &\geq n^4(n-1) \left(\frac{F'}{F} V^2 + \frac{F}{F'}{V'}^2\right),\\ \sum_{i=0}^n \sum_{j=0}^n (F'-2F_j')(F-2F_i)\| A_iA_j'\| ^2 &\geq n^4(n-1)^2\left(\frac{F'}{F} V^2 + \frac{F}{F'}{V'}^2\right),\\ \sum_{i=0}^n \sum_{j=0}^n F_j'F_i \| A_iA_j'\| ^2 &\geq n^4 \left(\frac{F'}{F} V^2 + \frac{F}{F'}{V'}^2\right), \end{aligned} \] and equalities hold if \(\Omega\) and \(\Omega '\) are regular and their centers are coincident. They also obtain several applications from these inequalities. For instance they obtain some inequalities concerning a simplex and its polar simplex.