Publications

Let 1

Continuing the theme of the previous volumes, these seminar notes reflect general trends in the study of Geometric Aspects of Functional Analysis, understood in a broad sense. Two classical topics represented are the Concentration of Measure Phenomenon in the Local Theory of Banach Spaces, which has recently had triumphs in Random Matrix Theory, and the Central Limit Theorem, one of the earliest examples of regularity and order in high dimensions. Central to the text is the study of the Poincaré and log-Sobolev functional inequalities, their reverses, and other inequalities, in which a crucial role is often played by convexity assumptions such as Log-Concavity. The concept and properties of Entropy form an important subject, with Bourgain's slicing problem and its variants drawing much attention. Constructions related to Convexity Theory are proposed and revisited, as well as inequalities that go beyond the Brunn-Minkowski theory. One of the major current research directions addressed is the identification of lower-dimensional structures with remarkable properties in rather arbitrary high-dimensional objects. In addition to functional analytic results, connections to Computer Science and to Differential Geometry are also discussed.

We introduce complex generalizations of the classical Legendre transform, operating on Kähler metrics on a compact complex manifold. These Legendre transforms give explicit local isometric symmetries for the Mabuchi metric on the space of Kähler metrics around any real analytic Kähler metric, answering a question originating in Semmes' work.

The purpose of this note is to draw attention to a misleading remark in the introduction of [1]. In our discussion of Theorem 1.2 we make the following claim: “one may check that the constants λ1 and λ2 in Theorem 1.2 are harmless polynomial functions of D”. Although we believe this to be true, the statement does not follow from the arguments of the paper. Several modifications are needed to obtain the claim, which we will now describe.

We present a short proof of the Alexandrov-Fenchel inequalities, which mixes elementary algebraic properties and convexity properties of mixed volumes of polytopes.

Let f: C
ⁿ→ C
^k be a holomorphic function and set Z= f
^{- 1}(0). Assume that Z is non-empty. We prove that for any r> 0 , γn(Z+r)≥γn(E+r),where Z+ r is the Euclidean r-neighborhood of Z; γ
_n is the standard Gaussian measure in C
ⁿ, and E⊆ C
ⁿ is an (n- k) -dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin.

This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar.

The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, our method is based on the following observation: When the Ricci curvature is nonnegative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to a geodesic foliation that is orthogonal to the level sets of a Lipschitz function. The Monge mass transfer problem plays an important role in our analysis.

According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the “hyperbolic” toric Kähler–Einstein equation e Φ = detD 2Φ on proper convex cones. We prove a generalization of this theorem by showing that for every Φ solving this equation on a proper convex domain Ω the corresponding metric measure space (D 2Φ, e Φ dx) has a non-positive Bakry–Émery tensor. Modifying the Calabi computations we obtain this result by applying the tensorial maximum principle to the weighted Laplacian of the Bakry–Émery tensor. Our computations are carried out in a generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of the log-concave probability measures we prove a third-order uniform dimension-free apriori estimate in the spirit of the second-order Caffarelli contraction theorem, which has numerous applications in probability theory.

We discuss the spectrum phenomenon for Lipschitz functions on the infinite-dimensional torus. Suppose that f is a measurable, real-valued, Lipschitz function on the torus T∞. We prove that there exists a number a∈R with the following property: For any ε>0, there exists a parallel, infinite-dimensional subtorus M⊆T∞ such that the restriction of the function f−a to the subtorus M has an L∞(M)-norm of at most ε.

We prove a log-Sobolev inequality for a certain class of log-concave measures in high dimension. These are the probability measures supported on the unit cube [0, 1]ⁿ ⊂ ℝⁿ whose density takes the form exp(−ψ), where the function ψ is assumed to be convex (but not strictly convex) with bounded pure second derivatives. Our argument relies on a transportation-cost inequality á la Talagrand.

In the study of concentration properties of isotropic log-concave measures, it is often useful to first ensure that the measure has super-Gaussian marginals. To this end, a standard preprocessing step is to convolve with a Gaussian measure, but this has the disadvantage of destroying small-ball information. We propose an alternative preprocessing step for making the measure seem super-Gaussian, at least up to reasonably high moments, which does not suffer from this caveat: namely, convolving the measure with a random orthogonal image of itself. As an application of this "inner-thickening", we recover Paouris' small-ball estimates.

We unify and slightly improve several bounds on the isotropic constant of high-dimensional convex bodies; in particular, a linear dependence on the body's psi(2) constant is obtained. Along the way, we present some new bounds on the volume of L-p-centroid bodies and yet another equivalent formulation of Bourgain's hyperplane conjecture. Our method is a combination of the Lp-centroid body technique of Paouris and the logarithmic Laplace transform technique of the first named author. (C) 2011 Elsevier Inc. All rights reserved.

Let G _∞=(C _m ^d)∞ denote the graph whose set of vertices is {0,⋯,m - 1} ^d, where two distinct vertices are adjacent if and only if they are either equal or adjacent in the m-cycle C _m in each coordinate. Let G _∞=(C _m ^d)∞ denote the graph on the same set of vertices in which two vertices are adjacent if and only if they are adjacent in one coordinate in C _m and equal in all others. Both graphs can be viewed as graphs of the d-dimensional torus. We prove that one can delete O(√dm ^d-1) vertices of G ₁ so that no topologically nontrivial cycles remain. This improves an O(d log ₂ (3/2)m ^d-1) estimate of Bollobás, Kindler, Leader and O'Donnell. We also give a short proof of a result implicit in a recent paper of Raz: one can delete an O(√d/m) fraction of the edges of G _∞ so that no topologically nontrivial cycles remain in this graph. Our technique also yields a short proof of a recent result of Kindler, O'Donnell, Rao and Wigderson; there is a subset of the continuous d-dimensional torus of surface area O(√d) that intersects all nontrivial cycles. All proofs are based on the same general idea: the consideration of random shifts of a body with small boundary and no nontrivial cycles, whose existence is proved by applying the isoperimetric inequality of Cheeger or its vertex or edge discrete analogues.

We present a counter-example to a certain conjecture that is related to Whitney's extension problems.

We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such "2-convex" bodies; in particular, the isotropic position is a finite volume-ratio position for these bodies. Second, we prove that high dimensional 2-convex bodies posses one-dimensional marginals that are approximately Gaussian. Third, we improve the known bounds on the isotropic constant of quotients of subspaces of L (p) and S-p(m), the Schatten Class space, for 1

We show that there exists a sequence εn↘0 for which the following holds: Let K⊂ℝⁿ be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝⁿ, t₀∈ℝ and σ>0 such that supA⊂R∣∣∣Prob{⟨X,θ⟩∈A}−12πσ−−−√∫Ae−(t−t0)22σ2dt∣∣∣≤εn,(∗) where the supremum runs over all measurable sets A⊂ℝ, and where 〈·,·〉 denotes the usual scalar product in ℝⁿ.
Furthermore, under the additional assumptions that the expectation of X
is zero and that the covariance matrix of X is the identity matrix, we
may assert that most unit vectors θ satisfy (*), with t₀=0 and σ=1. Corresponding principles also hold for multi-dimensional marginal distributions of convex sets.

Given an arbitrary set E⊂Rn, n≥2, and a function f:E→R, consider the problem of extending f to a C1 function defined on the entire Rn. A procedure for determining whether such an extension exists was suggested in 1958 by G. Glaeser. In 2004 C. Fefferman proposed a related procedure for dealing with the much more difficult cases of higher smoothness. The procedures in question require iterated computations of some bundles until the bundles stabilize. How many iterations are needed? We give a sharp estimate for the number of iterations that could be required in the C1 case. Some related questions are discussed.

In this note, we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same L-2 norm. We present several geometric applications of this result, the most important of which is a sharpening of the Johnson-Lindenstrauss embedding Lemma. Our results apply to a large class of random matrices, as we only require that the matrix entries have a subgaussian tail.

It is a classical fact, that given an arbitrary n-dimensional convex body, there exists an appropriate sequence of Minkowski symmetrizations (or Steiner symmetrizations), that converges in Hausdorff metric to a Euclidean ball. Here we provide quantitative estimates regarding this convergence, for both Minkowski and Steiner symmetrizations. Our estimates are polynomial in the dimension and in the logarithm of the desired distance to a Euclidean ball, improving previously known exponential estimates. Inspired by a method of Diaconis, our technique involves spherical harmonics. We also make use of an earlier result by the author regarding "isomorphic Minkowski symmetrization''.

We investigate the effect of a Steiner type symmetrization on the isotropic constant of a convex body. We reduce the problem of bounding the isotropic constant of an arbitrary convex body, to the problem of bounding the isotropic constant of a finite volume ratio body. We also add two observations concerning the slicing problem. The first is the equivalence of the problem to a reverse Brunn-Minkowski inequality in isotropic position. The second is the essential monotonicity in n of L-n = sup(Ksubset ofR)(n) L-K where the supremum is taken over all convex bodies in R-n, and L-K is the isotropic constant of K.