This is an announcement for the paper "Dvoretzky type theorems for multivariate polynomials and sections of convex bodies" by V.L. Dolnikov and R.N. Karasev.
Abstract: In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as the Birch theorem) and complex polynomials. We also consider a stronger conjecture on the homogeneous polynomial fields in the canonical bundle over real and complex Grassmannians. The latter conjecture is much stronger and false in general, but it is proved in the cases of $d=2$ (for $k$'s of certain type), odd $d$, the complex Grassmannian (for odd and even $d$ and any $k$). Corollaries for the John ellipsoid of projections or sections of a convex body are deduced from the case $d=2$ of the polynomial field conjecture.
Archive classification: math.MG math.AT math.CO math.FA
Mathematics Subject Classification: 46B20, 05D10, 26C10, 52A21, 52A23, 55M35
Submitted from: r_n_karasev@mail.ru
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1009.0392
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