Moscow--Petersburg seminar on low-dimensional mathematics SPb, PDMI, 30.11.2001, 16:00--17:45, room 311 A.S.Gasparyan (Pereslavl-Zalessky) Hyperdeterminants and a generalization of A.D.Aleksandrov's inequalities The famous inequality for mixed discriminants due to A.D.Alexandrov has determinantal nature for three reasons. Firstly, the mixed discriminant of the system of quadratic forms coincides (up to a factorial coefficient) with the hyperdeterminant of a cubic matrix. Secondly, Alexandrov's inequality expresses the nonnegativity of a symmetric determinant of second order. And thirdly, it is closely related to hyperbolicity of the form of determinant. The talk is devoted to a series of multidimensional-determinantal inequalities being generalizations of the Alexandrov inequality for mixed discriminants of quadratic forms and, in particular case, of the Yegorychev inequality for permanents. Applied to convex compact sets we have new inequalities for mixed volumes and for m-diameters of convex bodies that generalize the Alexandrov-Fenchel and Minkovsky inequalities. ----------------------------------------------------------- Home page of the seminar http://www.pdmi.ras.ru/~duzhin/LDM/