A characterization of the mixed discriminant

The striking analogy between mixed volumes of convex bodies and mixed discriminants of positive semidefinite matrices has repeatedly been observed. It was A. D. Aleksandrov [1] who, in his second proof of the Aleksandrov–Fenchel inequalities for the mixed volumes, first introduced the mixed discriminants of positive semidefinite quadratic forms and established some of their properties, including quadratic inequalities. This use of mixed discriminants in the theory of mixed volumes is described, for example, by Busemann [3, Sect. 7] and Leichtweiß [5]. In [7], the mixed volume of centrally symmetric convex bodies in Rn was characterized, up to a factor, as the only function of n centrally symmetric convex bodies which is Minkowski additive and increasing (with respect to set inclusion) in each variable and which vanishes if two of its arguments are parallel segments. The strong analogy mentioned above leads one to expect a similar characterization of the mixed discriminant, but it appears that the arguments employed in [7] cannot be transferred directly. The present note utilizes the L2 addition of ellipsoids with center at the origin, represented by symmetric positive semidefinite matrices. The L2 addition is a special case of the Lp addition (p ≥ 1) of convex bodies, which was introduced by Firey [4] and developed with great success by Lutwak, beginning with [6]. It turned out that the L2 addition opens the way to employ geometric arguments similar to those used in [7], which allows us to obtain the desired characterization. It is formulated as Theorem 2 below.