Some inequalities involving medians of two simplexes (Q1581018)

Let \(\Delta\) and \(\Delta'\) be two triangles with areas \(A\) and \(A'\) and lengths \(a_i\) and \(a_i'\), \(i=1,2,3\). In 1943, D. Pedoe proved the inequality \[ \sum_{i=1}^3 a_i^2\Biggl(\sum_{j=1}^3 a_j'{^2} - 2 a_i'{^2}\Biggr) \geq 16 AA', \] where equality holds if and only if \(\Delta\) is similar to \(\Delta'\). This result has been extended to \(n\)-dimensional simplexes by Yang Lu and Zhang Jingzhong and by Su Hauming. The author proves a similar generalization for \(n\)-dimensional simplexes \(\Omega\) and \(\Omega'\), \(n \geq 3\). For arbitrary real numbers \(0 < \alpha,\theta \leq 1\) the inequality \[ \sum_{i=1}^{n+1} m_i^{2\alpha}\Biggl(\sum_{j=1}^{n+1} m_j'{^{2\theta}} - 2 am_i'{^{2\theta}}\Biggr) \geq (n^2-1)\left[{(n+1)^{n-1}(n!)^2\over n^n}\right]^{(\alpha+\theta)/n} (V^\alpha V'{^\theta)^{2/n}} \] holds with equality if both \(\Omega\) and \(\Omega'\) are regular simplexes. In the formula, \(V\) and \(V'\) denote the volumes and \(m_i\), \(m_i'\) denote the medians of \(\Omega\) and \(\Omega'\) from vertex \(a_i\) and \(a_i'\), respectively.