To the theory of mixed volumes of convex bodies. Part III: Extension of two Minkowski theoremson convex polyhedra to all convex bodies

E X T E N SIO N OF TW O M IN K O W SK I T H EO R EM S ON C O N V E X PO LYH ED R A TO ALL C O N V E X BO DIES

In this paper we extend two results that were reported by Minkowski in his work Allgemeine Lebrsitze fiber die konvexen Polyeder,J namely, the theorem on the minimal property of polyhedra circumscribed about the ball and the theorem on the existence and uniqueness of a convex polyhedron with given directions and areas of faces. In the proofs we literally fol­ low Minkowski, and this, I hope, underlines the elegance and power of his method.