The dimension of almost spherical sections of convex bodies

Abstract

The well-known theorem of Dvoretzky [1] states that convex bodies of high dimension have low dimensional sections which are almost spherical. More precisely, the theorem states that for every integer k and every e > 0 there is an integer n(k, e) such that any Banach space X with dimension > n(k, e) has a subspace Y of dimension k with d{Y,l\) < 1 + e. Here d (Y, l£) In other words (considering the dependence of n(k, e) on k for fixed e) the dimension of the almost spherical section (of the unit ball) given by Dvoretzky's theorem is about the log of the dimension of the space. This estimate is in general the best possible, since as observed in [10] it is easy to verify that if X = Z~ any subspace Y of X whose Banach Mazur distance from a Hubert space is < 2 say, must be of dimension at most C log n. It turns out however that if we take into account not only X but also X much better information can be obtained. THEOREM 1. There is an absolute constant c > 0 so that for every Banach space X of dimension n the following formula holdsThis formula needs explanation. The number k(X) is an integer with the following property: "most" subspaces Y of X of dimension k(X) satisfy d (Y, l£(x)) ^ 2. We do not intend to make the word "most" precise here. What really matters is that there exists aFCl with d{Y, l^Xy) < 2. The number \\P\\ is the norm of a projection P. If k(X) < k(X ), P is a projection from X onto a suitable subspace Y of X with d(Y, 1%( X )) ^ 2 (again actually "most" subspaces Y of dimension k(X) work). If k(X*) < k(X), P is a projection from X onto a subspace Z C X with d(Z 9 l^( X *)) ^ 2. (1970). Primary 52A20, 52A25, 46C05, 46E30. AMS (MOS) subject classifications

Spaces of Operators, the ψ-Daugavet Property, and Numerical Indices

“…If N has atomic type I n M-summands for n < ∞, then there exists a finite rank projection P : N → N for which I N − P = 1. This establishes (4). It remains to prove (3).…”

“…Then |f j (e i )| = 1 for any i and j . By [4], ln N · ln M K 2 n, where K is an absolute constant. Assume that N M, and prove that X contains an isometric copy of m 1 (the case of N < M is dealt with in the same way).…”

Logarithmic reduction of the level of randomness in some probabilistic geometric constructions

“…The fact that the cross polytope B( n 1 ) has sections of proportional dimension which are isomorphic to Euclidean has been known for long, see [12] for sections of small proportional dimension, in which case you get almost isometric embeddings, and [18] for (n/2)-dimensional sections isomorphic to Euclidean up to universal constants. The proofs show that a random section satisfies this with high probability.…”

“…One such example is a section of the cross-polytope (the unit ball of n 1 ) which is of proportional dimension and isomorphic to a Euclidean ball up to a constant independent of the dimension n. It has been known from the early works such as [12,18] that a random section, with high probability, satisfies this property (random here with respect to the Haar measure on the Grassmanian of all, say, (n/2)-dimensional subspaces of R n ). However, there is no known explicit construction of such a section of n 1 .…”

Quantitative Steiner/Schwarz-type symmetrizations

“…6 (1) ForallO < e <_ 1 andO < A <_ e, An >_ 1 there existlessthan cn2/E 2 log 1/e orthogonal pairs of (1 -A )n-symmetrizations that will change (1. 6.1) to:…”