Some geometric inequalities for a tangent point simplex (Q5946241)

The author derives some inequalities for \(d\)-dimensional simplices in Euclidean \(d\)-space where the following quantities are combined: volume of the considered simplex \(S\), volume of its ``tangent point simplex'' (= convex hull of the touching points of the insphere of \(S\)), volumes of regular simplices that, in certain ways, are metrically related to \(S\), circum- and inradius of \(S\). In form of the respective equality cases, these inequalities yield characterizations of regular simplices.