Two inequalities concerning the circumradius and inradius of a simplex in \(n\)-dimensional space (Q2735358)

Let \(\Delta_n\) be an \(n\)-dimensional simplex in the Euclidean \(n\)-space with vertices \(A_1,\dots,A_n\). Let \(O\), \(R\), \(I\), \(r\) and \(G\) denote the circumcenter, the circumradius, the incenter, the inradius and the barycenter of \(\Delta_n\), respectively. Let \(\alpha_{k,ij}\) denote the angle formed by the edges \(A_kA_i\) and \(A_kA_j\) and put \(\vartheta_{k,ijl} = (\alpha_{k,ij}+\alpha_{k,jl}+\alpha_{k,li}) / 2\) and \(f(n)=\prod_{\lambda=4}^n {\lambda \over \lambda^2 - 1}\). NEWLINENEWLINENEWLINEThe authors prove the following two inequalities: NEWLINE\[NEWLINER^2 \geq (\text{csc }\vartheta_{k,ijl})^{f(n)/4n}n^2r^2 + \overline{OI}^2\tag{1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINER^2 \geq (\text{csc }\vartheta_{k,ijl})^{f(n)/4n}n^2r^2 + \overline{OG}^2/4.\tag{2}NEWLINE\]NEWLINE In both cases, equality holds if and only if \(\Delta_n\) is regular. NEWLINENEWLINENEWLINEFor the proof the authors utilize barycentric coordinates. NEWLINENEWLINENEWLINEThese two inequalities improve the Euler inequality as well as another inequality given by \textit{M. S. Klamkin} [Inequality for a simplex, SIAM Rev. 27, 576 (1985)].