Geometric and Functional Analysis

, Volume 21, Issue 2, pp 301–318 | Cite as

Dvoretzky Type Theorems for Multivariate Polynomials and Sections of Convex Bodies

  • Vladimir L. Dol’nikov
  • Roman N. KarasevEmail author


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. This 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, and 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.

Keywords and phrases

Dvoretzky’s theorem Ramsey type theorems multivariate polynomials 

2010 Mathematics Subject Classification

46B20 05D10 26C10 52A21 52A23 55M35 


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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of AlgebraYaroslavl’ State UniversityYaroslavl’Russia
  2. 2.Dept. of MathematicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia

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