On the icosahedron inequality of László Fejes-Tóth

Abstract

Abstract. In this paper we deal with the problem of finding the maximal volume polyhedra with a prescribed property and inscribed in the unit sphere. We generalize an inequality (called icosahedron inequality) of L. Fejes-Tóth which has the following interesting consequence: the regular icosahedron has maximal volume in the class of the polyhedra having 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) simplicial polyhedra, whose number of faces, and also the list of the maximal edge lengths of the faces are given. As a consequence of this inequality we prove a conjecture which states that the maximal volume polyhedron spanned by the vertices of two regular simplices with common centroid is the cube.Mathematics subject classification (2010): 52A40, 52A38, 26B15.

Volume of Convex Hull of Two Bodies and Related Problems

“…We call again the tetrahedron ABCO the facial tetrahedron with base ABC and apex O. [22]) Let ABC be a triangle inscribed in the unit sphere. Then there is an isosceles triangle A ′ B ′ C ′ inscribed in the unit sphere with the following properties:…”

“…The following theorem gives an upper bound on the volume of the star-shaped polyhedron corresponding to the given spherical tiling in question. [22]) Assume that 0 < τ i < π/2 holds for all i.…”

“…In the paper [22] the author closed this problem using a generalization of the icosahedron inequality of L. Fejes-Tóth. It has been shown the general statement: Theorem 3.15.…”