The new \(k\)-\(n\)-type Neuberg-Pedoe inequalities (Q2518179)

The classical Neuberg-Pedoe inequality states a relation between the areas of two triangles \(\Delta_i\) and their edge-lengths \(a_i,b_i,c_i\), \(i=1,2\), namely, \[ 16A(\Delta_1)A(\Delta_2)\leq a_1^2(-a_2^2+b_2^2+c_2^2)+b_1^2(a_2^2-b_2^2+c_2^2)+c_1^2(a_2^2+b_2^2-c_2^2), \] with equality if and only if \(\Delta_1,\Delta_2\) are similar. In this paper the authors state several inequalities relating the volumes of two \(n\)-simplices with the \(k\)-dimensional volumes of their \(k\)-faces.