Improvements of the \(n\)-dimensional Euler inequality (Q1339798)

For an \(n\)-dimensional simplex in \(E^ n\) it is well known that \(R \geq nr\) (generalizing Euler's inequality \(R \geq 2r\) for a triangle), \textit{G. Korchmáros} [Atti Accad. Naz. Lincei, VIII. Ser., Rend. Cl. Sci. Fis. Mat. Nat. 56, 876-879 (1974; Zbl 0309.50007)] obtained an upper bound for the volume of an \(n\)-dimensional simplex. The authors of the present paper obtain a considerable refinement of this bound, and as a consequence they obtain an improvement of \(R \geq nr\) similar to the one obtained by \textit{M. S. Klamkin} [SIAM Rev. 27, 576 (1985)] namely \(R^ 2 \geq n^ 2 r^ 2 + OI^ 2\).