Leng, Gangsong; Ma, Tongyi; Qian, Xiangzheng Inequalities for a simplex and an interior point. (English) Zbl 0990.51009 Geom. Dedicata 85, No. 1-3, 1-10 (2001). Let \(S\) be an \(n\)-dimensional simplex in Euclidean \(n\)-space whose circum- and inradius are given by \(R\) and \(r\), respectively. Further, \(d_i\) be the distance from an interior point \(P\) of \(S\) to its \(i\)th facet, \(i=0,1, \dots, n\).The authors prove that \[ \sum_{0\leq i<j\leq n}{1 \over d_id_j} \geq{1\over r^2}+ {C_n\over R^2}, \] where \(C_n= {1\over 2}(n-1)n^2 (n+2)\) and equality holds iff \(S\) is regular with centroid \(P\). They even give a generalization of this result and a related statement referring also to the circumradii of the facets of \(S\). Finally they show that \[ {1\over |r-|OI||^\alpha}- {1\over r^\alpha} \geq{(n-1)n^2 \over R^\alpha} \] with equality iff \(S\) is regular, where \(\alpha\geq 1\) and \(O\) and \(I\) denote the circum- and incenter of \(S\), respectively. Reviewer: Horst Martini (Chemnitz) Cited in 5 Documents MSC: 51M16 Inequalities and extremum problems in real or complex geometry 52A40 Inequalities and extremum problems involving convexity in convex geometry 51M20 Polyhedra and polytopes; regular figures, division of spaces Keywords:regular simplex; circumradius; circumcenter; volume; simplex; inradius; incenter PDFBibTeX XMLCite \textit{G. Leng} et al., Geom. Dedicata 85, No. 1--3, 1--10 (2001; Zbl 0990.51009) Full Text: DOI