Publications
2020

(2020) Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 20172019 Volume II. Klartag B. & Milman E.(eds.). p. 4363 Abstract
We study the lower bound for Koldobsky's slicing inequality. We show that there exists a measure mu and a symmetric convex body K subset of Rn, such that for all xi is an element of S(n)1 and all t is an element of R,mu(+) (K boolean AND (xi(perpendicular to) + t xi))Our bound is optimal, up to the value of the universal constant. It improves slightly upon the results of the first named author and Koldobsky, which included a doublylogarithmic error. The proof is based on an efficient way of discretizing the unit sphere.

(2020) Revista Matematica Iberoamericana. Abstract
We present a coordinatefree version of Fefferman's solution of Whitney's extension problem in the space Cm−1,1(Rn). While the original argument relies on an elaborate induction on collections of partial derivatives, our proof uses the language of ideals and translationinvariant subspaces in the ring of polynomials. We emphasize the role of compactness in the proof, first in the familiar sense of topological compactness, but also in the sense of finiteness theorems arising in logic and semialgebraic geometry. In a followup paper, we apply these ideas to study extension problems for a class of subRiemannian manifolds where global coordinates may be unavailable.

(2020) American Journal of Mathematics. 142, 1, p. 323339 Abstract
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.

(2020) Journal of the European Mathematical Society. 22, 2, p. 477505 Abstract
The complex method of interpolation, going back to Calderón and Coifman et al., on the one hand, and the AlexanderWermerSlodkowski theorem on polynomial hulls with convex fibers, on the other hand, are generalized to a method of interpolation of real (finitedimensional) Banach spaces and of convex functions. The underlying duality in this method is given by the Legendre transform. Our results can also be interpreted as new properties of solutions of the homogeneous complex MongeAmpère equation.

(2020) 31st Annual ACMSIAM Symposium on Discrete Algorithms, SODA 2020. Chawla S.(eds.). p. 14961508 Abstract
The convex body chasing problem, introduced by Friedman and Linial [FL93], is a competitive analysis problem on any normed vector space. In convex body chasing, for each timestep t ∈ N, a convex body Kt ⊆ Rd is given as a request, and the player picks a point xt ∈ Kt. The player aims to ensure that the total distance moved PTt=0−1 xt−xt+1 is within a bounded ratio of the smallest possible offline solution. In this work, we consider the nested version of the problem, in which the sequence (Kt) must be decreasing. For Euclidean spaces, we consider a memoryless algorithm which moves to the socalled Steiner point, and show that in an appropriate sense it is exactly optimal among memoryless algorithms. For general finite dimensional normed spaces, we combine the Steiner point and our recent algorithm in [ABC+19] to obtain a new algorithm which is nearly optimal for all `pd spaces with p ≥ 1, closing a polynomial gap.
2019

(2019) Geometriae Dedicata. Abstract
Let M be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by σ. Let f:M→R be a nonconstant eigenfunction of the Laplacian. The random wave conjecture suggests that in certain situations, the value distribution of f under σ is approximately Gaussian. Write μ for the measure whose density with respect to σ is ∇f2. We observe that the value distribution of f under μ admits a unimodal density attaining its maximum at the origin. Thus, in a sense, the zero set of an eigenfunction is the largest of all level sets. When M is a manifold with boundary, the same holds for Laplace eigenfunctions satisfying either the Dirichlet or the Neumann boundary conditions. Additionally, we prove a monotonicity formula for level sets of solid spherical harmonics, essentially by viewing nodal sets of harmonic functions as weighted minimal hypersurfaces.

(2019) Comptes Rendus Mathematique. 357, 8, p. 676680 Abstract
We present a short proof of the AlexandrovFenchel inequalities, which mixes elementary algebraic properties and convexity properties of mixed volumes of polytopes.

(2019) Journal of Geometric Analysis. 29, 3, p. 23472373 Abstract
Given a convex body K⊂Rn with the barycenter at the origin, we consider the corresponding Kähler–Einstein equation e−Φ=detD2Φ. If K is a simplex, then the Ricci tensor of the Hessian metric D2Φ is constant and equals n−14(n+1). We conjecture that the Ricci tensor of D2Φ for an arbitrary convex body K⊆Rn is uniformly bounded from above by n−14(n+1) and we verify this conjecture in the twodimensional case. The general case remains open.

(2019) Studia Mathematica. 247, 1, p. 85107 Abstract
We discuss interplays between logconcave functions and logconcave sequences. We prove a Bernsteintype theorem, which characterizes the Laplace transform of logconcave measures on the halfline in terms of logconcavity of the alternating Taylor coefficients. We establish concavity inequalities for sequences inspired by the PrekopaLeindler and the Walkup theorems. One of our main tools is a stochastic variational formula for the Poisson average.
2018

(2018) Proceedings of the American Mathematical Society. 146, 11, p. 48794888 Abstract
For p ≥ 1, n ∈ ℕ, and an originsymmetric convex body K in ℝ n, let (formula presented) be the outer volume ratio distance from K to the class L n p of the unit balls of ndimensional subspaces of L p. We prove that there exists an absolute constant c > 0 such that (formula presented) This result follows from a new slicing inequality for arbitrary measures, in the spirit of the slicing problem of Bourgain. Namely, there exists an absolute constant C > 0 so that for any p ≥ 1, any n ∈ ℕ, any compact set K ⊆ ℝ n of positive volume,∫ and any Borel measurable function f ≥ 0 on K, (formula presented) where the supremum is taken over all affine hyperplanes H in ℝ n. Combining the above display with a recent counterexample for the slicing problem with arbitrary measures from the work of the second and third authors [J. Funct. Anal. 274 (2018), pp. 2089–2112], we get the lower estimate from the first display. In turn, the second inequality follows from an estimate for the pth absolute moments of the function f (formula presented). Finally, we prove a result of the BusemannPetty type for these moments.

(2018) Journal of Geometric Analysis. 28, 3, p. 20082027 Abstract
Let f: C
n→ C
k be a holomorphic function and set Z= f
 1(0). Assume that Z is nonempty. We prove that for any r> 0 , γn(Z+r)≥γn(E+r),where Z+ r is the Euclidean rneighborhood 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. 
(2018) Advances in Mathematics. 330, p. 74108 Abstract
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 centrallysymmetric. 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 counterexample to a conjecture by Kuperberg on the distribution of volume in a body and in its polar.

(2018) Journal of Functional Analysis. 274, 7, p. 20892112 Abstract
For n∈N, let S
n be the smallest number S>0 satisfying the inequality ∫Kf≤S⋅K
[Formula presented]⋅maxξ∈S
n−1∫K∩ξ
⊥f for all centrallysymmetric convex bodies K in R
n and all even, continuous probability densities f on K. Here K is the volume of K. It was proved in [16] that S
n≤2n, and in analogy with Bourgain's slicing problem, it was asked whether S
n is bounded from above by a universal constant. In this note we construct an example showing that S
n≥cn/loglogn, where c>0 is an absolute constant. Additionally, for any 0<α<2 we describe a related example that satisfies the socalled ψ
αcondition. 
(2018) St. Petersburg Mathematical Journal. 29, 1, p. 107138 Abstract
We find that for any ndimensional, compact, convex set K ⊆ R n+1 there is an affinelyspherical hypersurface M ⊆ R n+1 with center in the relative interior of K such that the disjoint union M ∪ K is the boundary of an (n + 1) dimensional, compact, convex set. This socalled affine hemisphere M is uniquely determined by K up to affine transformations, it is of elliptic type, is associated with K in an affinelyinvariant manner, and it is centered at the Santaló point of K.
2017

(2017) Proceedings 58th Annual IEEE Symposium on Foundations of Computer Science. p. 639650 (Annual IEEE Symposium on Foundations of Computer Science). Abstract
Let X be a set of n points of norm at most 1 in the Euclidean space R
k
, and suppose ε > 0. An εdistance sketch for X is a data structure
that, given any two points of X enables one to recover the square of the
(Euclidean) distance between them up to an additive error of ε. Let
f(n, k, ε) denote the minimum possible number of bits of such a sketch.
Here we determine f(n, k, ε) up to a constant factor for all n ≥ k ≥ 1
and all ε ≥ 1/n
0.49
. Our proof is algorithmic, and provides an efficient algorithm for
computing a sketch of size O(f(n, k, ε)/n) for each point, so that the
square of the distance between any two points can be computed from their
sketches up to an additive error of ε in time linear in the length of
the sketches. We also discuss the case of smaller ε > 2/√n and obtain
some new results about dimension reduction in this range. In
particular, we show that for any such ε and any k ≤ t = log(2+ε
2
n)/ε
2
there are configurations of n points in R
k
that cannot be embedded in R
ℓ
for ℓ <; ck with c a small absolute positive constant, without
distorting some inner products (and distances) by more than ε. On the
positive side, we provide a randomized polynomial time algorithm for a
bipartite variant of the JohnsonLindenstrauss lemma in which scalar
products are approximated up to an additive error of at most ε. This
variant allows a reduction of the dimension down to O(log(2+ε
2
n)/ε
2
), where n is the number of points. 
(2017) Memoirs of the American Mathematical Society. Vol. 249(1180). p. 177 Abstract
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, logconcave 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.

(2017) Analysis Mathematica. 43, 1, p. 6788 Abstract
According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a nonpositive 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 nonpositive 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 logconcave probability measures we prove a thirdorder uniform dimensionfree apriori estimate in the spirit of the secondorder Caffarelli contraction theorem, which has numerous applications in probability theory.

(2017) Geometric and Functional Analysis. 27, 1, p. 130164 Abstract
This paper presents connections between Gromov’s work on isoperimetry of waists and Milman’s work on the Mellipsoid of a convex body. It is proven that any convex body K⊆Rn has a linear image K~⊆Rn of volume one satisfying the following waist inequality: Any continuous map f:K~→Rℓ has a fiber f−1(t) whose (n−ℓ)dimensional volume is at least cn−ℓ, where c>0 is a universal constant. In the specific case where K=[0,1]n it is shown that one may take K~=K and c=1,
confirming a conjecture by Guth. We furthermore exhibit relations
between waist inequalities and various geometric characteristics of the
convex body K. 
(2017) Geometric Aspects of Functional Analysis. Klartag B. & Milman E.(eds.). Cham, Switzerland: . Vol. 2169. p. 187211 Abstract
We establish the following universality
property in high dimensions: Let X be a random vector with
density in RnRn. The density function can
be arbitrary. We show that there exists a fixed unit vector θ∈Rnθ∈Rn such that the random variable Y=⟨X,θ⟩Y=⟨X,θ⟩ satisfiesmin{P(Y≥tM),P(Y≤−tM)}≥ce−Ct2for all 0≤t≤c~n−−√,min{P(Y≥tM),P(Y≤−tM)}≥ce−Ct2for
all 0≤t≤c~n,where M > 0 is any median of  Y  , i.e., min{P(Y≥M),P(Y≤M)}≥1/2min{P(Y≥M),P(Y≤M)}≥1/2. Here, c,c~,C>0c,c~,C>0 are universal constants. The dependence on the
dimension n is optimal, up to universal constants, improving
upon our previous work.
2016

(2016) Canadian Journal Of MathematicsJournal Canadien De Mathematiques. 68, 3, p. 655674 Abstract
We study a natural discrete Bochnertype inequality on graphs, and explore its merit as a notion of "curvature" in discrete spaces. An appealing feature of this discrete version of the socalled Gamma(2)calculus (of BakryEmery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest, particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Busertype inequalities (a la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edgeisoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs, a result of independent interest.

(2016) Communications in Contemporary Mathematics. 18, 1, 1550029. Abstract
We discuss the spectrum phenomenon for Lipschitz functions on the infinitedimensional torus. Suppose that f is a measurable, realvalued, 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, infinitedimensional 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.

Needle decompositions in Riemannian geometry(2016)
in Mem. Amer. Math. Soc. Here are related slides.

Discrete curvature and abelian groups(2016) Canad. J. Math.. Vol. 68 (3), p. 655–674.
2015

(2015) Journal of Functional Analysis. 268, 12, p. 38343866 Abstract
With any convex function ψ on a finitedimensional linear space X such that ψ goes to +∞ at infinity , we associate a Borel measure μ on X*. The measure μ is obtained by pushing forward the measure e(ψ(x)) dx under the differential of ψ. We propose a class of convex functions  the essentiallycontinuous, convex functions  for which the above correspondence is in fact a bijection onto the class of finite Borel measures whose barycenter is at the origin and whose support spans X*. The construction is related to toric KahlerEinstein metrics in complex geometry, to Prekopa's inequality, and to the Minkowski problem in convex geometry.

(2015) Analysis and PDE. 8, 1, p. 3355 Abstract
We investigate the Brenier map del Phi between the uniform measures on two convex domains in Rn, or, more generally, between two logconcave probability measures on Rn. We show that the eigenvalues of the Hessian matrix D2 Phi exhibit concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension n.
2014

(2014) Annali della Scuola normale superiore di Pisa  Classe di scienze. 13, 4, p. 9751007 Abstract
We prove stability estimates for the BrunnMinkowski inequality for convex sets. As opposed to previous stability results, our estimates improve as the dimension grows. In particular, we obtain a nontrivial conclusion for high dimensions already when (Equation Presented) Our results are equivalent to a thin shell bound, which is one of the central ingredients in the proof of the central limit theorem for convex sets.

(2014) Israel Journal of Mathematics. 203, 1, p. 5980 Abstract
We prove a logSobolev inequality for a certain class of logconcave 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 transportationcost inequality á la Talagrand.

(2014) Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics. Klartag B. & Milman E.(eds.). Switzerland: . Vol. 2116. p. 231260 Abstract
We discuss a certain Riemannian metric, related to the toric KählerEinstein equation, that is associated in a linearlyinvariant manner with a given logconcave measure in Rn. We use this metric in order to bound the second derivatives of the solution to the toric KählerEinstein equation, and in order to obtain spectralgap estimates similar to those of Payne and Weinberger.

(2014) Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics. Klartag B. & Milman E.(eds.). Switzerland: . Vol. 2116. p. 123131 Abstract
Given an arbitrary 1Lipschitz function f on the torus Tn, we find a kdimensional subtorus M ⊆ Tn, parallel to the axes, such that the restriction of f to the subtorus M is nearly a constant function. The kdimensional subtorus M is selected randomly and uniformly. We show that when k ≤ c log n/(log log n + log 1/ε), the maximum and the minimum of f on this random subtorus M differ by at most ε, with high probability.
2013

(2013) Annales de la Faculté des sciences de Toulouse. 12, 1, p. 141 Abstract
We propose a new method for obtaining Poincaretype inequalities on arbitrary convex bodies in R^n. Our technique involves a dual version of Bochner's formula and a certain moment map, and it also applies to some nonconvex sets. In particular, we generalize the central limit theorem for convex bodies to a class of nonconvex domains, including the unit balls of L_pspaces in R^n for 0 < p < 1.
2012

(2012) Journal of Functional Analysis. 262, 9, p. 41814204 Abstract
We give a functional version of the affine isoperimetric inequality for logconcave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincaré inequality for the Gaussian measure.

(2012) Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics. Klartag B., Mendelson S. & Milman VD.(eds.). Berlin: . Vol. 2050. p. 267278 Abstract
In the study of concentration properties of isotropic logconcave measures, it is often useful to first ensure that the measure has superGaussian marginals. To this end, a standard preprocessing step is to convolve with a Gaussian measure, but this has the disadvantage of destroying smallball information. We propose an alternative preprocessing step for making the measure seem superGaussian, 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 "innerthickening", we recover Paouris' smallball estimates.

(2012) Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics. Klartag B., Mendelson S. & Milman V. D.(eds.). Berlin: . Vol. 2050. p. 151168 Abstract
We survey some interplays between spectral estimates of Hörmandertype,
degenerate MongeAmpère equations and geometric inequalities related to
logconcavity such as BrunnMinkowski, Santaló or Busemann inequalities. 
(2012) Journal of Functional Analysis. 262, 1, p. 1034 Abstract
We unify and slightly improve several bounds on the isotropic constant of highdimensional 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 Lpcentroid bodies and yet another equivalent formulation of Bourgain's hyperplane conjecture. Our method is a combination of the Lpcentroid body technique of Paouris and the logarithmic Laplace transform technique of the first named author. (C) 2011 Elsevier Inc. All rights reserved.

(2012) Theory of Probability and its Applications. 56, 3, p. 403419 Abstract
Suppose that X1,. . ., Xn are independent, identically distributed random variables of mean zero and variance one. Assume that E vertical bar X1 vertical bar(4) 0 is a universal constant. This inequality should be compared with the classical Berry Esseen theorem, according to which the left hand side may decay with n at the slower rate of O(1/root n) for the unit vector 0  (1,. . ., 1)/root n. An explicit, universal example for coefficients theta = (theta(1),. . ., theta(n)) for which this inequality holds is theta = ( 1,root 2, 1, root 2, 1, root 2, 1, root 2,. . .) (3n/2)(1/2), when n is divisible by four. Parts of the argument are applicable also in the more general case, in which X1,. . ., Xn are independent random variables of mean zero and variance one yet are not necessarily identically distributed. In this general setting, the bound above holds with delta(4) = n(1) Sigma(n)(j)=1 E vertical bar Xj vertical bar(4) for most selections of a unit vector theta = (theta(1),. . ., theta(n)) is an element of Rn. Here "most" refers to the uniform probability measure on the unit sphere.
2011

(2011) STOC'11. p. 3140 Abstract
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, boundederror) protocol requires poly(n) bits of communication. That quantum protocol requires two rounds of communication. Ever since Raz's 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.

(2011) Concentration, Functional Inequalities and Isoperimetry. Houdre C., Milman E., Ledoux M. & Milman M.(eds.). p. 5568 Abstract
We discuss connections between certain wellknown open problems related to the uniform measure on a highdimensional convex body. In particular, we show that the "thin shell conjecture" implies the "hyperplane conjecture". This extends a result by K. Ball, according to which the stronger "spectral gap conjecture" implies the "hyperplane conjecture".
2010

(2010) Journal of the European Mathematical Society. 12, 3, p. 723754 Abstract
Suppose that μ is an absolutely continuous probability measure on ℝn, for large n. Then μ has lowdimensional marginals that are approximately sphericallysymmetric. More precisely, if n ≥ (C/epse;)Cd, then there exist ddimensional marginals of μ that are εfar from being spherically symmetric, in an appropriate sense. Here C > 0 is a universal constant.

(2010) Proceedings of the Fifth European Congress of Mathematics. Ran A., te Riele H. & Wiegerinck J.(eds.). Switzerland: . p. 401417 Abstract
We review recent advances in the understanding of probability measures withgeometric characteristics on Rn, for largen. These advances include the centrallimit theorem for convex sets, according to which the uniform measure on a highdimensional convex body has marginals that are approximately gaussian.
2009

(2009) Probability Theory and Related Fields. 145, 12, p. 133 Abstract
Suppose X = (X 1, . . . , X n ) is a random vector, distributed uniformly in a convex body K ⊂ ℝn. We assume the normalization EXi2 = 1 for i = 1, . . . , n. The body K is further required to be invariant under coordinate reflections, that is, we assume that (±X 1, . . . , ±X n ) has the same distribution as (X 1, . . . , X n ) for any choice of signs. Then, we show that E (X  n,)2 ≤ Cn where C ≤ 4 is a positive universal constant, and  •  is the standard Euclidean norm in ℝn. The estimate is tight, up to the value of the constant. It leads to a BerryEsseen type bound in the central limit theorem for unconditional convex bodies.

(2009) Journal of Topology and Analysis. 1, 2, p. 101111 Abstract
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 mcycle 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 ddimensional torus. We prove that one can delete O(√dm d1) vertices of G 1 so that no topologically nontrivial cycles remain. This improves an O(d log 2 (3/2)m d1) 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 ddimensional 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.

(2009) Israel Journal of Mathematics. 170, 1, p. 253268 Abstract
Let N a parts per thousand yen n + 1, and denote by K the convex hull of N independent standard gaussian random vectors in R(n). We prove that with high probability, the isotropic constant of K is bounded by a universal constant. Thus we verify the hyperplane conjecture for the class of gaussian random polytopes.

(2009) Annals of Mathematics. 169, 1, p. 315346 Abstract
Suppose we are given a finite subset E⊂Rn and a function f:E→R. How to extend f to a Cm function F:Rn→R with Cm norm of the smallest possible order of magnitude? In this paper and in [20] we tackle this question from the perspective of theoretical computer science. We exhibit algorithms for constructing such an extension function F, and for computing the order of magnitude of its Cm norm. The running time of our algorithms is never more than CNlogN, where N is the cardinality of E and C is a constant depending only on m and n.

(2009) Revista Matematica Iberoamericana. 25, 1, p. 49–273 Abstract
We exhibit efficient algorithms to perform the following task: Given a function f defined on a finite subset E⊂Rn, compute a Cm function F on Rn, with a controlled Cm norm, that approximates f on the subset E.

(2009) Revista Matematica Iberoamericana. 25, 2, p. 423446 Abstract
We present a counterexample to a certain conjecture that is related to Whitney's extension problems.
2008

(2008) Journal of Functional Analysis. 254, 8, p. 22752293 Abstract
We prove a pointwise version of the multidimensional central limit theorem for convex bodies. Namely, let μ be an isotropic, logconcave probability measure on Rn. For a typical subspace E ⊂ Rn of dimension nc, consider the probability density of the projection of μ onto E. We show that the ratio between this probability density and the standard Gaussian density in E is very close to 1 in large parts of E. Here c > 0 is a universal constant. This complements a recent result by the second named author, where the total variation metric between the densities was considered.

(2008) Israel Journal of Mathematics. 164, 1, p. 221249 Abstract
We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such "2convex" bodies; in particular, the isotropic position is a finite volumeratio position for these bodies. Second, we prove that high dimensional 2convex bodies posses onedimensional marginals that are approximately Gaussian. Third, we improve the known bounds on the isotropic constant of quotients of subspaces of L (p) and Sp(m), the Schatten Class space, for 1 <p ≤ 2.

(2008) St. Petersburg Mathematical Journal. 19, 1, p. 77106 Abstract
A wellknown consequence of the Brunn–Minkowski inequality says that the distribution of a linear functional on a convex set has a uniformly subexponential tail. That is, for any dimension n, any convex set K ⊂ ℝn of volume one, and any linear functional φ: ℝn →ℝ, we have Voln((x ∈ K; φ(x) t φ L1(K))) ≤ e−ct for all t 1, where φ L1(K) = ∫K φ(x) dx and c 0 is a universal constant. In this paper, it is proved that for any dimension n and a convex set K ⊂ ℝn of volume one, there exists a nonzero linear functional φ: ℝn → ℝ such that (Formula Presented), where c 0 is a universal constant.
2007

(2007) Journal of Functional Analysis. 245, 1, p. 284310 Abstract
We investigate the rate of convergence in the central limit theorem for convex sets established in [B. Klartag, A central limit theorem for convex sets, Invent. Math., in press. [8]]. We obtain bounds with a powerlaw dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets.

(2007) Inventiones Mathematicae. 168, 1, p. 91131 Abstract
We show that there exists a sequence εn↘0 for which the following holds: Let K⊂ℝn be a compact, convex set with a nonempty 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 multidimensional marginal distributions of convex sets. 
(2007) Geometric Aspects of Functional Analysis. Schechtman G.. & Milman V. D.(eds.). Berlin Heidelberg: . Vol. 1910. p. 133166 Abstract
This note consists of three parts. In the first, we observe that a surprisingly rich family of functional inequalities may be proven from the Brunn–Minkowskiinequality using a simple geometric technique. In the second part, we discuss consequences of a functional version of Santal ́o’s inequality, and in the third partwe consider functional counterparts of mixed volumes and of Alexandrov–Fenchelinequalities.

(2007) Revista Matematica Iberoamericana. 23, 2, p. 635669 Abstract
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.

(2007) Israel Journal of Mathematics. 157, 1, p. 193207 Abstract
Large deviation estimates are by now a standard tool in Asymptotic Convex Geometry, contrary to small deviation results. In this note we present a novel application of a small deviations inequality to a problem that is related to the diameters of random sections of high dimensional convex bodies. Our results imply an unexpected distinction between the lower and upper inclusions in Dvoretzky's Theorem.
2006

(2006) Geometric and Functional Analysis. 16, 6, p. 12741290 Abstract
Let K ⊂ ℝn be a convex body and ε >0. We prove the existence of another convex body K' ⊂ ℝn, whose BanachMazur distance from K is bounded by 1+ε, such that the isotropic constant of K' is smaller than c \√ ε, where c > 0 is a universal constant. As an application of our result, we present a slight improvement on the best general upper bound for the isotropic constant, due to Bourgain. © Birkhäuser Verlag, Basel 2006.

(2006) Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006. Soria J., Verdera J., Varona J. L. & SanzSolé M.(eds.). Vol. II. p. 15471562 Abstract
The classical theorems of highdimensional convex geometry exhibit a surprising level of regularity and order in arbitrary highdimensional convex sets. These theorems are mainly concerned with the rough geometric features of general convex sets; the socalled "isomorphic" features. Recent results indicate that, perhaps, highdimensional convex sets are also very regular on the almostisometric scale. We review some related research directions in highdimensional convex geometry, focusing in particular on the problem of geometric symmetrization.
2005

(2005) Combinatorics Probability and Computing. 14, 56, p. 829843 Abstract
For an arbitrary ndimensional convex body, at least almost n Steiner symmetrizations are required in order to symmetrize the body into an isomorphic ellipsoid. We say that a body T ⊂ ℝn is 'quickly symmetrizable with function $c(\varepsilon)$' if for any $\varepsilon > 0$ there exist only $\lfloor \varepsilon n \rfloor$ symmetrizations that transform T into a body which is $c(\varepsilon)$isomorphic to an ellipsoid. In this note we ask, given a body K ⊂ ℝn, whether it is possible to remove a small portion of its volume and obtain a body $T \subset K$ which is quickly symmetrizable. We show that this question, for $c(\varepsilon) $ polynomially depending on 1\ε, is equivalent to the slicing problem.

(2005) Journal of Functional Analysis. 225, 1, p. 229245 Abstract
In this note, we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same L2 norm. We present several geometric applications of this result, the most important of which is a sharpening of the JohnsonLindenstrauss embedding Lemma. Our results apply to a large class of random matrices, as we only require that the matrix entries have a subgaussian tail.

(2005) Geometriae Dedicata. 112, 1, p. 169182 Abstract
We present a view of logconcave measures, which enables one to build an isomorphic theory for high dimensional logconcave measures, analogous to the corresponding theory for convex bodies. Concepts such as duality and the Minkowski sum are described for logconcave functions. In this context, we interpret the BrunnMinkowski and the BlaschkeSantaló inequalities and prove the two corresponding reverse inequalities. We also prove an analog of Milman's quotient of subspace theorem, and present a functional version of the Urysohn inequality.

(2005) Journal of Functional Analysis. 218, 2, p. 372394 Abstract
Here we show that any centrallysymmetric convex body K⊂ ℝn has a perturbation T⊂ ℝn which is convex and centrallysymmetric, such that the isotropic constant of T is universally bounded. T is close to K in the sense that the BanachMazur distance between T and K is O(log n). If K is a body of a nontrivial type then the distance is universally bounded. The distance is also universally bounded if the perturbation T is allowed to be nonconvex. Our technique involves the use of mixed volumes and AlexandrovFenchel inequalities. Some additional applications of this technique are presented here.
2004

(2004) Mathematika. 51, 12, p. 3348 Abstract
Let ℒ(f) denote the Legendre transform of a function f: ℝn → 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 f̃(x) = f(x  a) such that ∫ℝne f̃ ∫ℝneℒ(f̃)≤ (2π)n. (1) If a is selected so as to minimize the left side of (1), then equality in (1) is satisfied if and only if ef is proportional to the distribution of a Gaussian random variable. This inequality immediately implies the Santaló inequality for convex bodies, as well as a new concentration inequality for the Gaussian measure.

(2004) Geometric and Functional Analysis. 14, 6, p. 1322 – 1338 Abstract
It is a classical fact, that given an arbitrary ndimensional 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''.

(2004) Proceedings of the American Mathematical Society. 132, 9, p. 26192628 Abstract
We present an integral inequality connecting volumes and diameters of sections of a convex body. We apply this inequality to obtain some new inequalities concerning diameters of sections of convex bodies, among which is our "low Mestimate". Also, we give novel, alternative proofs to some known results, such as the fact that a finite volume ratio body has proportional sections that are isomorphic to a Euclidean ball.

(2004) Geometric Aspects of Functional Analysis. Schechtman G. & Milman VD.(eds.). Berlin Heidelberg: . Vol. 1850. p. 101115 Abstract
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 BrunnMinkowski inequality in isotropic position. The second is the essential monotonicity in n of Ln = sup(Ksubset ofR)(n) LK where the supremum is taken over all convex bodies in Rn, and LK is the isotropic constant of K.

(2004) Geometric Aspects of Functional Analysis. Schechtman G. & Milman V. D.(eds.). Berlin Heidelberg: . Vol. 1850. p. 149158 Abstract
Given an arbitrary convex symmetric ndimensional body, we construct a natural and nontrivial continuous map which associates ellipsoids to ellipsoids, such that the LownerJohn ellipsoid of K is its unique fixed point. A new characterization of the LownerJohn ellipsoid is obtained, and we also gain information regarding the contact points of inscribed ellipsoids with K.
2003

(2003) Inventiones Mathematicae. 153, 3, p. 463485 Abstract
This paper proves that there exist 3n Steiner symmetrizations that transform any convex set K ⊂ ℝn into an isomorphic Euclidean ball; i.e. if vol(K) = vol(Dn) where Dn is the standard Euclidean unit ball, then K can be transformed into a body K̃ such that c1Dn ⊂ K̃ ⊂ c2D n, where c1, c2 are numerical constants. Moreover, for any c > 2, cn symmetrizations are also enough.

(2003) Comptes Rendus Mathematique. 336, 4, p. 331334 Abstract
Here we discuss results around the slicing problem, which is a well known open problem in asymptotic convex geometry. We show that if one can prove that the isotropic constant of bodies with a finite volume ratio is uniformly bounded  then it would follow that the isotropic constant of any convex body is uniformly bounded.
2002

(2002) Annals of Mathematics. 156, 3, p. 947960 Abstract
This paper proves that for every convex body in ℝn there exist 5n Minkowski symmetrizations which transform the body into an approximate Euclidean ball. This result complements the sharp cn log n upper estimate by J. Bourgain, J. Lindenstrauss and V.D. Milman, of the number of random Minkowski symmetrizations sufficient for approaching an approximate Euclidean ball.
2000

(2000) Geometric Aspects of Functional Analysis. Schechtman G. & Milman V. D.(eds.). Berlin, Heidelberg: . Vol. 1745. p. 109117 Abstract
Here we extend a result by J. Bourgain, J. Lindenstrauss, V.D. Milman on the number of random Minkowski symmetrizations needed to obtain an approximated ball, if we start from an arbitrary convex body in ℝn. We also show that the number of “deterministic” symmetrizations needed to approximate an Euclidean ball may be significantly smaller than the number of “random” ones.