On the icosahedron inequality of L\'aszl\'o Fejes-T\'oth

In this paper we deal with the problem to find the maximal volume polyhedra with a prescribed property and inscribed in the unit sphere. We generalize those inequality (called by \emph{icosahedron inequality}) of L. Fejes-T\'oth of which an interesting consequence the fact that regular icosahedron has maximal volume in the class of the polyhedra with twelve vertices and inscribed in the unit sphere. We give an upper bound for the volume of such star-shaped (with respect to the origin) triangular polyhedra which we know the maximal edge lengths of its faces, respectively. As a further consequence of the generalized inequality we prove the conjecture which states that the maximal volume polyhedron spanned by the vertices of two regular simplices with common centroid is the cube.