A inequality for two simplices and two points (Q1766356)

Let \(S\), \(S'\) be two \(n\)-dimensional simplices, with \(A_0,A_1,\dots, A_n\) as vertex set, \(V\) as volume and \(F_i\) as facet (opposite to \(A_i\)) of \(S\). For \(S'\) we use analogous notations, e.g. \(V'\) for its volume. Let \(P\) be an arbitrary point of Euclidean \(n\)-space, and \(P'\) be an interior point of \(S'\). The authors prove the following: If \(D_i=|PA_i|\) and \(d_i\) denotes the distance from \(P'\) to the \(i\)th facet of \(S'\), then \[ \sum^n_{i=0} {D^2_i\over d_i}\geq C(n)\biggl({V^2\over V'}\biggr)^{{1\over n}} \] with \(C(n)= n^{{3\over 2}}(n!)^{{1\over n}}(n+ 1)^{{(n-1)\over 2n}}\), where equality holds if \(S\) and \(S'\) are regular with \(P\) and \(P'\) as circumcenter, respectively.