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.
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Home page of the seminar http://www.pdmi.ras.ru/~duzhin/LDM/