Inequalities for edge lengths and volumes of two simplexes (Q1376483)

If two triangles \(A_iB_iC_i\) \((i=1,2)\) have side lengths \(a_i\), \(b_i\), \(c_i\) and areas \(F_i\), then \[ a^2_1(b^2_2 +c^2_2- a^2_2)+ a^2_2(b^2_1 +c^2_1- a^2_1)\geq 16F_1 F_2, \] with equality if and only if the triangles are similar. This is the Nirenberg-Pedoe inequality. The author mentions two sharpenings of this inequality, one of which was proved by \textit{P. Chia-Kuei} in Crux. Math. 10, 68-69 (1984)]. He then generalizes these last two inequalities to \(n\)-dimensional space. As a result, he obtains a generalization of the Neuberg-Pedoe inequality, which is different from the generalization obtained by \textit{Yang Lu} and \textit{Zhang Jing Zhong} in Bull. Aust. Math. Soc. 27, 203-214 (1983; Zbl 0519.51013).