Inequalities involving the inradii of simplexes
Let \(r\) be the inradius of an \(n\)-simplex \(A = A_1 A_2 \dots A_{n+1}\) in \(E^n\) \((n \geq 3)\), and \(r_i\) the inradius of the \((n-1)\)-simplex \(F_i = A_1 \dots A_{i-1} A_{i+1} \dots A_{n+1}\). The author proves that \(\sum^{n+1}_{j=1} r^{-2}_j \leq (n-1)r^{-2}\), with equality if and only if \(A\) is regular. He also proves an inequality concerning the inradii of \(k\)-simplexes and \(\ell\)-simplexes \((3 \leq k < \ell \leq n)\) whose vertices lie in a finite set of points in \(E^n\).
scientific article; zbMATH DE number 872724