Inequalities for any point and two simplices (Q1301711)

Let \(\Omega\) and \(\Omega'\) be two simplices in \(\mathbb{R}^n\), and let \(P\) be a point in \(\Omega\). Let \(d_i\) denote the distance of \(P\) to the facets \(F_i\) of \(\Omega\), and let \(h_i\) denote the altitude of \(\Omega'\) from the vertex \(A_i'\) (of \(\Omega')\). The authors derive several sharp upper bounds for certain expressions containing all the \(d_i\) and the \(h_i\) where the bounds contain the \(n\)-dimensional volumes of \(\Omega,\Omega'\), respectively. For example they show that \[ \sum^n_{i=0} (d_0\cdots \widehat d_i\cdots d_n)(h_0 \cdots \widehat h_i\cdots h_n)\leq \biggl[\bigl((n-1)! \bigr)^2/n^{n-2}\biggr] VV', \] where \(\widehat d_i\) \((\widehat h_i)\) means that this factor is omitted in the product.