Bo'az Klartag

A central limit theorem for convex sets

We show that there exists a sequence

The Santaló point of a function, and a functional form of the Santaló inequality

\varepsilon_n\searrow0

Quantum one-way communication can be exponentially stronger than classical communication

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

Uniform almost sub-Gaussian estimates for linear functionals on convex sets

\sup_{A\subset\mathbb{R}}\left|\textit{Prob}\,\{\langle X,\theta\rangle\in A\}-\frac{1}{\sqrt{2\pi\sigma}}\int_Ae^{-\frac{(t - t_0)^2}{2\sigma^2}} dt\right|\leq\varepsilon_n,\qquad{(\ast)}

Fitting a

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.

C^m

Let L(f) denote the Legendre transform of a function f : ℝ n → ℝ. A theorem of K. Ball about even functions is generalized, and it is proved that, for any measurable function f ≥ 0, there exists a translation f(x) = f(x−a) such that

-Smooth Function to Data II

In STOC 1999, Raz presented a (partial) function for which there is a quantum protocol communicating only O(log n) qubits, but for which any classical (randomized, bounded-error) protocol requires poly(n) bits of communication. That quantum protocol requires two rounds of communication. Ever since Razs paper it was open whether the same exponential separation can be achieved with a quantum protocol that uses only one round of communication. Here we settle this question in the affirmative.

Needle Decompositions in Riemannian Geometry

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 Open image in new window where the supremum is taken over all convex bodies in Open image in new window , and L K is the isotropic constant of K.