Generalization and sharpening of Safta's conjecture in the \(n\)-dimensional space (Q1581020)

The author proves the following statements: In an \(n\)-dimensional simplex \(S\subset E^n\) with vertex set \(\{A_1,\dots, A_{n+1}\}\) and \(P\) as an interior point, let \(B_i\) denote the intersection point of the line \(g(A_iP)\) and the facet hyperplane which is opposite to \(A_i\), \(i=1,\dots, n+1\). Furthermore, \(C_i\) be the intersection point of \(g(A_iP)\) and the hyperplane \(aff(B_1, \dots, B_{i-1},B_{i+1}, \dots,B_{n+1})\). The the inequalities \[ \prod^{n+1} _{i=1}{A_iC_i\over C_iB_i}\geq(n-1)^{n+1}\quad \text{and} \quad \prod^{n+1}_{i=1} {A_iB_i\over C_iB_i}\geq n^{n+1} \] hold, with equality iff \(P\) is the centroid of the simplex \(S\).