ABSTRACT
M IX E D D IS C R IM IN A N T S A N D M IX E D V O L U M E S
This paper is devoted, first of all, to studying the invariants of a fam ily of quadratic forms in n variables, the so-called mixed discriminants of these forms. The relation between these discriminants and the questions in the theory of convex bodies was established first by Minkowski and Weyl. 2 The algebraic part (§ 1-3) of the work is presented without any reference to convex bodies since it possibly has intrinsic interest. Furthermore, I demonstrate how one can, basing on the algebraic results, deduce the main theorems of the theory of mixed volumes of convex bodies, true, confining oneself only to bodies with twice differentiable support functions. In § 4 I derive the concept of mixed volume and its properties from an application of mixed discriminants. In §5 I prove the uniqueness of a convex body with given curvature functions without taking recourse to the inequalities between mixed volumes. In § 6 I deduce the main inequality between mixed volumes by the Hilbert method .3 Finally, in § 7 I generalize two theorems proved in Part II, namely, the theorem on the uniqueness of a convex body with given curvature functions and the theorem on the equality in the Brunn inequality. This last section stands somewhat alone and is rather closely re lated to Part II of this work.
