Inequalities for medians of two simplices (Q1433244)

The authors prove the following inequality: Let \(\Omega\) and \(\Omega'\) be two \(n\)-simplices with volume \(V\) and \(V'\) and with medians \(m_0, \dots, m_n\) and \(m'_0, \dots , m'_n\) respectively. Let \(M= \sum_{i=0}^{n} m_{i}^{2}\) and \(M'= \sum_{i=0}^{n} m_{i}^{'2}\). Then \[ \sum _{i=0}^{n} m_{i}^{'2}(M-\lambda_n m_{i}^{2})\geq \frac 12 \mu_n\Bigl(\frac{n!^2}{n+1}\Bigr)^{2/n}\Bigl(\frac{M'}{M}V^{4/n}+ \frac{M}{M'}V^{'4/n}\Bigr), \] where \(\lambda_n=\frac{n(n+1)}{n(n-1)+1}\) and \(\mu_n= \frac{(n+1)^4(n-1)}{n^2(n(n-1)+1)}\). For \(n>2\) equality holds if and only if \(\Omega\) and \(\Omega'\) are regular; for \(n=2\), equality holds if and only if \(\Omega\) and \(\Omega'\) are similar. Reviewer's remark: There is a misprint in page 109, line 1. It says: Applying Theorem 1\dots. It should say: Applying Lemma 3. I think also that the problem is not motivated.