Metric Entropy of Homogeneous Spaces

For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em covering number} $N(K , \varepsilon )$ is defined as the smallest number of balls of radius $\varepsilon$ whose union covers $K$. Knowledge of the {\em metric entropy}, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous spaces of unitary (or orthogonal) groups with respect to some natural metrics, most notably the one induced by the operator norm. This generalizes earlier author's results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. In the process we give a characterization of geodesics in $U(n)$ (or $SO(m)$) for a class of non-Riemannian metric structures

Confidence regions for means of multivariate normal distributions and a non-symmetric correlation inequality for gaussian measure

Let $\mu$ be a Gaussian measure (say, on ${\bf R}^n$) and let $K, L \subset {\bf R}^n$ be such that K is convex, $L$ is a "layer" (i.e. $L = \{x : a \leq < x,u > \leq b \}$ for some $a$, $b \in {\bf R}$ and $u \in {\bf R}^n$) and the centers of mass (with respect to $\mu$) of $K$ and $L$ coincide. Then $\mu(K \cap L) \geq \mu(K) \cdot \mu(L)$. This is motivated by the well-known "positive correlation conjecture" for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses an apparently hitherto unknown estimate for the (standard) Gaussian cumulative distribution function: $\Phi (x) > 1 - \frac{(8/\pi)^{{1/2}}}{3x + (x^2 +8)^{{1/2}}} e^{-x^2/2}$ (valid for $x > -1$)

Saturating Constructions for Normed Spaces II

We prove several results of the following type: given finite dimensional normed space V possessing certain geometric property there exists another space X having the same property and such that (1) log (dim X) = O(log (dim V)) and (2) every subspace of X, whose dimension is not "too small," contains a further well-complemented subspace nearly isometric to V. This sheds new light on the structure of large subspaces or quotients of normed spaces (resp., large sections or linear images of convex bodies) and provides definitive solutions to several problems stated in the 1980s by V. Milman. The proofs are probabilistic and depend on careful analysis of images of convex sets under Gaussian linear maps.Comment: 35 p., LATEX; the paper is a follow up on math.FA/040723