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Inequalities for a simplex and an interior point. (English) Zbl 0990.51009

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.

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
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