Dvoretzky type theorems for multivariate polynomials and sections of convex bodies. (English) Zbl 1232.46043

Dol’nikov, Vladimir L.; Karasev, Roman N.

Geom. Funct. Anal. 21, No. 2, 301-318 (2011).

The main result of this interesting paper is the following “Dvoretzky-type” result: For any \(d, k \in \mathbb N\) with \(d\) even, there is \(n(d,k) \in \mathbb N\) such that for every \(n \geq n(d,k)\) and for every homogeneous polynomial \(f:\mathbb R^n \to \mathbb R\) of degree \(d,\) there is a subspace \(V \subset \mathbb R^n, \;\dim\;V = k,\) such that \(f|_V\) is proportional to \((x_1^2 + ... + x_n^2)^{d/2}.\) This result is known as the Gromov-Milman conjecture [V. D. Milman, Lect. Notes Math. 1317, 283–289 (1988; Zbl 0657.10020)]. (The original conjecture also stated that \(n(d,k)\) is of the order \(k^d;\) the authors note that their proof, which uses the Borsuk-Ulam theorem, does not settle this.) When \(d\) is odd the result can be found in [B. J. Birch, Mathematika, Lond. 4, 102–105 (1957; Zbl 0081.04501)], although there is more recent work as well as estimates on \(n(d,k)\) by P. Hájek and the reviewer in [Arch. Math. 86, No. 6, 561–568 (2006; Zbl 1106.46029)]. In fact, the estimates provided in the previous reference are improved here. Among other things, the authors show (Theorem 5) that for any \(d, k, m \in \mathbb N\) with \(d\) odd, there is \(n(d,k,m) \in \mathbb N\) such that for any \(n \geq n(d,k,m)\) and any odd polynomial map \(f:\mathbb R^n \to \mathbb R^m,\) all of whose coordinate functions have degree at most \(d,\) there is a \(k\)-dimensional subspace of \(\mathbb R^n\) that \(f\) maps to \(0.\) An explicit value of \(n(d,k,m)\) is given, as is a complex version of this result (Theorem 7) with a better bound.
The authors discuss a stronger conjecture involving the Grassmannian of subspaces of \(\mathbb R^n\) of dimension \(k\). Although this conjecture is known to be false in general, the authors show that it does hold in certain cases such as \(k = 2p^\alpha\) where \(p\) is a prime number. As a consequence, they prove that for such \(k\) and all \(m,\) the following holds for sufficiently large \(n\): For any \(m\) convex bodies \(K_1, \dots, K_m\) in \(\mathbb R^n,\) there is a \(k\)-dimensional subspace \(L \subset \mathbb R^n\) such that the orthogonal projections of each \(K_i\) onto \(L\) has a Euclidean ball as its John ellipsoid.

Reviewer: Richard M. Aron (Kent)

MSC:

46G25	(Spaces of) multilinear mappings, polynomials
46B07	Local theory of Banach spaces
05D10	Ramsey theory
52A21	Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry)
55M35	Finite groups of transformations in algebraic topology (including Smith theory)

Keywords:

Dvoretzky’s theorem; Ramsey type theorems; multivariate polynomials; John ellipsoid; Gromov-Milman conjecture

Citations:

Zbl 0657.10020; Zbl 0081.04501; Zbl 1106.46029

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