Research Interests

1. Convex Geometry

2. Geometric Functional Analysis

3. and applications of Probability and Harmonic Analysis to the above subjects.

Please, check below for the  list of my papers/surveys/lecture notes with abstracts, please, also check the work done by my graduate students.

Lecture notes, Surveys ...

• Analytic methods in convex geometry. (with Dmitry Ryabogin), Lectures given at the Polish Academy of Sciences 21 - 26, November 2011, to appear in IMPAN Lecture Notes.

The notion of duality is a very important notions in mathematics. It is playing one of the central roles in Functional Analysis and Convex Geometry. It is hard to believe, but there are still a number of “classical” open problems tightly connected to this notion. Our main goal here is to discuss some of these problems and to present just a few of the classical and some modern analytic methods that might be useful to attack them. We will talk about Mahler's conjecture on the minimality of the product of volumes of a convex body and its polar. We will also discuss some problem of the local characterizations of some classes of convex bodies (zonoids). Finally, we will present a number of open problems connected with the uniqueness (or non-uniqueness!) of convex bodies given by the volumes of their projections or/and sections. In particular, we will talk about recent solutions of the problems of Klee and Bonensen.

• "Harmonic Analysis and Uniqueness questions in Convex Geometry." (with Dmitry Ryabogin and Vladyslav Yaskin), to appear in Advances in Classic and Applied Analysis, on the occasion of Konstantin Oskolkov's 65th Birthday.

In this paper we discuss some open questions on the unique determination of convex bodies from the volume of sections and projections.

• "Unified Fourier analytic approach to the volume of projections and sections of convex bodies," (with Alexander Koldobsky and Dmitry Ryabogin). Fourier Analysis and Convexity, (Editors: L. Brandolini, L. Cozani, A. Iosevich and G. Travaglini), Birkhauser 2004, 119-131.

It has been noticed long ago that many results on sections and projections are dual to each other, though methods used in the proofs are quite different and don't use the duality of underlying structures directly. The goal of this survey is to bring together certain aspects of the Fourier approaches to sections and projections, in order to emphasize the similarities between the results and the proofs.

Research papers

• "Polytopes of Maximal Volume Product," (with Matthew Alexander and Matthieu Fradelizi), submitted.

For a convex body $K \subset {\mathbb R}^n$, let $K^z = \{y\in{\mathbb R}^n : \langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$ be the polar body of $K$ with respect to the center of polarity $z \in {\mathbb R}^n$. The goal of this paper is to study the maximum of the volume product $\mathcal{P}(K)=\min_{z\in {\rm int}(K)}|K||K^z|$, among convex polytopes $K\subset {\mathbb R}^n$ with a number of vertices bounded by some fixed integer $m \ge n+1$. In particular, we prove that the supremum is reached at a simplicial polytope with exactly $m$ vertices and we provide a new proof of a result of Meyer and Reisner showing that, in the plane, the regular polygon has maximal volume product among all polygons with at most $m$ vertices. Finally, we treat the case of polytopes with $n+2$ vertices in ${\mathbb R}^n$.

• Let us define for a compact set $A \subset {\mathbb R}^n$ the sequence $$A(k) = \left\{\frac{a_1+\cdots +a_k}{k}: a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big).$$ It was independently proved by Shapley, Folkman and Starr (1969) and by Emerson and Greenleaf (1969) that $A(k)$ approaches the convex hull of $A$ in the Hausdorff distance induced by the Euclidean norm as $k$ goes to $\infty$. We explore in this survey how exactly $A(k)$ approaches the convex hull of $A$, and more generally, how a Minkowski sum of possibly different compact sets approaches convexity, as measured by various indices of non-convexity. The non-convexity indices considered include the Hausdorff distance induced by any norm on ${\mathbb R}^n$, the volume deficit (the difference of volumes), a non-convexity index introduced by Schneider (1975), and the effective standard deviation or inner radius. After first clarifying the interrelationships between these various indices of non-convexity, which were previously either unknown or scattered in the literature, we show that the volume deficit of $A(k)$ does not monotonically decrease to 0 in dimension 12 or above, thus falsifying a conjecture of Bobkov et al. (2011), even though their conjecture is proved to be true in dimension 1 and for certain sets $A$ with special structure. On the other hand, Schneider's index possesses a strong monotonicity property along the sequence $A(k)$, and both the Hausdorff distance and effective standard deviation are eventually monotone (once $k$ exceeds $n$). Along the way, we obtain new inequalities for the volume of the Minkowski sum of compact sets (showing that this is fractionally superadditive but not supermodular in general, but is indeed supermodular when the sets are convex), demonstrate applications of our results to combinatorial discrepancy theory, and suggest some questions worthy of further investigation.

• In this paper we present a series of inequalities connecting the surface area measure of a convex body and surface area measure of its projections and sections. We present a solution of a question of S. Campi, P. Gritzmann and P. Gronchi regarding the asymptotic behavior of the best constant in a recently proposed reverse Loomis-Whitney inequality. Next we give a new sufficient condition for the slicing problem to have an affirmative answer, in terms of the least outer volume ratio distance'' from the class of intersection bodies of projections of at least proportional dimension of convex bodies. Finally, we show that certain geometric quantities such as the volume ratio and minimal surface area (after a suitable normalization) are not necessarily close to each other.

• "Distribution functions of sections and projections of convex bodies," (with Jaegil Kim and Vladyslav Yaskin), J. London Math. Soc., to appear.

Typically, when we are given the section (or projection) function of a convex body, it means that in each direction we know the size of the central section (or projection) perpendicular to this direction. Suppose now that we can only get the information about the sizes of sections (or projections), and not about the corresponding directions. In this paper we study to what extent the distribution function of the areas of central sections (or projections) of a convex body can be used to derive some information about the body, its volume, etc.

• "Characterization of simplices via the Bezout inequality for mixed volumes." (with Christos Saroglou and Ivan Soprunov), Proc. Amer. Math. Soc., to appear.

We consider the following Bezout inequality for mixed volumes: $$V(K_1,\dots,K_r,\Delta[{n-r}])V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(K_i,\Delta[{n-1}])\ \text{ for }2\leq r\leq n.$$ It was shown previously that the inequality is true for any $n$-dimensional simplex $\Delta$ and any convex bodies $K_1, \dots, K_r$ in ${\mathbb R}^n$. It was conjectured that simplices are the only convex bodies for which the inequality holds for arbitrary bodies $K_1, \dots, K_r$ in ${\mathbb R}^n$. In this paper we prove that this is indeed the case if we assume that $\Delta$ is a convex polytope. Thus the Bezout inequality characterizes simplices in the class of convex $n$-polytopes. In addition, we show that if a body $\Delta$ satisfies the Bezout inequality for all bodies $K_1, \dots, K_r$ then the boundary of $\Delta$ cannot have strict points. In particular, it cannot have points with positive Gaussian curvature.

• "Iterations of the projection body operator and a remark on Petty's conjectured projection inequality." (with Christos Saroglou), Journal of Functional Analysis, to appear.

We prove that if a convex body has absolutely continuous surface area measure, whose density is sufficiently close to the constant, then the sequence $\{\Pi^m K\}$ of convex bodies converges to the ball with respect to the Banach-Mazur distance, as $m\rightarrow\infty$. Here, $\Pi$ denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty and to improve a related inequality of Lutwak.

• "Do Minkowski averages get progressively more convex?" (with Matthieu Fradelizi Mokshay Madiman and Arnaud Marsiglietti), C. R. Acad. Sci. Paris, Ser. I, 354 (2016) 185-189.

Let us define, for a compact set $A \subset {\mathbb R}^n$, the Minkowski averages of $A$: $$A(k) = \left\{\frac{a_1+\cdots +a_k}{k} : a_1, \ldots, a_k\in A\right\}=\frac{1}{k}\Big(\underset{k\ {\rm times}}{\underbrace{A + \cdots + A}}\Big).$$ We study monotonicity properties of $A(k)$ towards convexity when considering the volume deficit and a non-convexity index of Schneider. For the volume deficit, we show that monotonicity fails in general, thus disproving a conjecture of Bobkov, Madiman and Wang. For Schneider's non-convexity index, we show that a strong form of monotonicity holds.

• "A discrete version of Koldobsky's slicing inequality," (with Matthiew Alexander and Marthin Henk), Israel J. Math., to appear.

Let $\# K$ be a number of integer lattice points contained in a set $K$. In this paper we prove that for each $d\in {\mathbb N}$ there exists a constant $C(d)$ depending on $d$ only, such that for any origin-symmetric convex body $K \subset {\mathbb R}^d$ containing $d$ linearly independent lattice points $$\# K \leq C(d)\max(\# (K\cap H)) {\rm vol}_d(K)^{\frac{d-m}{d}},$$ where the maximum is taken over all $m$-dimensional subspaces of ${\mathbb R}^d$. We also prove that $C(d)$ can be chosen asymptotically of order $O(1)^{d}d^{d-m}$. In addition, we show that if $K$ is an unconditional convex body then $C(d)$ can be chosen asymptotically of order $O(d)^{d-m}$.

• "Bezout Inequality for Mixed volumes," (with Ivan Soprunov), to appear in International Mathematics Research Notices.

In this work we consider the following analog of Bezout inequality for mixed volumes: $$V(P_1,\dots,P_r,\Delta^{n-r})V_n(\Delta)^{r-1}\leq \prod_{i=1}^r V(P_i,\Delta^{n-1})\ \text{ for }2\leq r\leq n.$$ We show that the above inequality is true when $\Delta$ is an $n$-dimensional simplex and $P_1, \dots, P_r$ are convex bodies in ${\mathbb R}^n$. We conjecture that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $\Delta$ must be an $n$-dimensional simplex. We prove that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $\Delta$ must be indecomposable (i.e. cannot be written as the Minkowski sum of two convex bodies which are not homothetic to $\Delta$), which confirms the conjecture when $\Delta$ is a simple polytope and in the $2$-dimensional case. Finally, we connect the inequality to an inequality on the volume of orthogonal projections of convex bodies as well as prove an isomorphic version of the inequality.

• "On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalitiess," (with Galyna Livshyts, Arnaud Marsiglietti and Piotr Nayar), Trans. Amer. Math. Soc., to appear.

In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality $$\mu(\lambda A + (1-\lambda)B)^{1/n} \geq \lambda \mu(A)^{1/n} + (1-\lambda)\mu(B)^{1/n}$$ holds true for an unconditional product measure $\mu$ with decreasing density and a pair of unconditional convex bodies $A,B \subset {\mathbb R}^n$. We also show that the above inequality is true for any unconditional $\log$-concave measure $\mu$ and unconditional convex bodies $A,B \subset {\mathbb R}^n$. Finally, we prove that the inequality is true for a symmetric $\log$-concave measure $\mu$ and a pair of symmetric convex sets $A,B \subset {\mathbb R}^2$, which, in particular, settles two-dimensional case of the conjecture for Gaussian measure proposed by A.Z. and R. Gardenr (see the paper below). In addition, we deduce the $1/n$-concavity of the parallel volume $t \mapsto \mu(A+tB)$, Brunn's type theorem and certain analogues of Minkowski first inequality.

• "An isomorphic version of the Busemann-Petty problem for arbitrary measures," (with Alexander Koldobsky ), Geometriae Dedicata, Volume 174, Issue 1 (2015), 261-277.

The Busemann-Petty problem for an arbitrary measure $\mu$ with non-negative even continuous density in ${\mathbb R}^n$ asks whether origin-symmetric convex bodies in ${\mathbb R}^n$ with smaller $(n-1)$-dimensional measure $\mu$ of all central hyperplane sections necessarily have smaller measure $\mu.$ It was shown by the second author that the answer to this problem is affirmative for $n\le 4$ and negative for $n\ge 5$. In this paper we prove an isomorphic version of this result. Namely, if $K,M$ are origin-symmetric convex bodies in ${\mathbb R}^n$ such that $\mu(K\cap \xi^\bot)\le \mu(M\cap \xi^\bot)$ for every $\xi \in {\mathbb S}^{n-1},$ then $\mu(K)\le \sqrt{n}\ \mu(M).$ Here $\xi^\bot$ is the central hyperplane perpendicular to $\xi.$ We also study the above question with additional assumptions on the body $K$ and present the complex version of the problem. In the special case where the measure $\mu$ is convex we show that $\sqrt{n}$ can be replaced by $cL_n,$ where $L_n$ is the maximal isotropic constant. Note that, by a recent result of Klartag, $L_n \le O(n^{1/4})$. Finally we prove a slicing inequality $$\mu(K)\le C n^{1/4} \max_{\xi \in {\mathbb S}^{n-1}} \mu(K \cap \xi^\perp)\ {\rm vol}_n(K)^{\frac 1n}$$ for any convex even measure $\mu$ and any symmetric convex body $K$ in ${\mathbb R}^n,$ where $C$ is an absolute constant. This inequality was recently proved by the first author for arbitrary measures with continuous density, but with $\sqrt{n}$ in place of $n^{1/4}.$

• "Stability of the reverse Blaschke-Santalo inequality for unconditional convex bodies," (with Jaegil Kim), Proc. Amer. Math. Soc. 143 (2015), 1705-1717.

Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in ${\mathbb R^n}$. The corresponding inequality to the conjecture is sometimes called the the reverse Blaschke-Santalo inequality. The conjecture is known in ${\mathbb R}^2$ and in several special cases. In the class of unconditional convex bodies, Saint Raymond confirmed the conjecture, and Meyer and Reisner, independently, characterized the equality case. In this paper we present a stability version of these results and also show that any symmetric convex body, which is sufficiently close to an unconditional body, satisfies the the reverse Blaschke-Santalo inequality.

• "A Remark on the Extremal Non-Central Sections of the Unit Cube," (with James Moody, Corey Stone and David Zach), "Asymptotic Geometric Analysis", Fields Institute Communications 221-229, 68, M. Ludwig et al. (eds.), Springer Science+Business Media, New York 2013.

In this paper, we investigate extremal volumes of non-central slices of the unit cube. The case of central hyperplane sections is known and was studied in Hadwiger, Ball, Hensley and others. The case of non-central sections, i.e. when we dictate that the hyperplane must be a certain distance t>0 from the center of the cube, is open in general and the same is true about sections of the unit cube by slabs, with some partial results provided by Barthe and Kolodbsky and by Koenig and Koldobsky. In this paper we give a full solution for extremal one-dimensional sections and a partial solution for extremal hyperplane slices for the case $t >\frac{\sqrt{n-1}}{2}$. We also make a remark on minimal volume slices of the cube by slabs of width 2t, when $t >\frac{\sqrt{n-1}}{2}$.

• "An asymmetric convex body with maximal sections of constant volume," (with Fedor Nazarov and Dmitry Ryabogin), Journal of AMS, 27 (2014), no. 1, 43–68.

We show that in all dimensions $d>2$, there exists an asymmetric convex body of revolution all of whose maximal hyperplane sections have the same volume. This gives the negative answer to the question posed by V. Klee in 1969.

• Non-uniqueness of convex bodies with prescribed volumes of sections and projections. (with Fedor Nazarov and Dmitry Ryabogin), Mathematika, 59 (2013), 213-221.

We show that if $d\ge 4$ is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.

• "An application of shadow systems to Mahler's conjecture," (with Matthieu Fradelizi, Mathieu Meyer). Discrete & Computationa Geometry, 48 (2012), no 3, 721-734.

We elaborate on the use of shadow systems to prove particular case of the conjectured lower bound of the volume product $\mathcal{P}(K)=\min_{z\in {\rm int}(K)}|K|||K^z|$ in ${\mathbb R^n}$. In particular, we show that if $K\subset {\mathbb R}^3$ is the convex hull of two $2$-dimensional convex bodies, then $\mathcal{P}(K) \ge \mathcal{P}(\Delta^3)$, where $\Delta^3$ is an $3$-dimensional simplex, thus confirming the $3$-dimensional case of Mahler conjecture, for this class of bodies. A similar result is provided for symmetric case, where we prove that if $K\subset {\mathbb R}^3$ is symmetric and the convex hull of two $2$-dimensional convex bodies, then $\mathcal{P}(K) \ge \mathcal{P}(B_\infty^3)$, where $B_\infty^3$ is the unit cube.

• "A problem of Klee on inner section functions of convex body," (with Richar J. Gardner, Dmitry Ryabogin, Vladyslav Yaskin), J. Differential Geometry, 91 (2012), 261-279.

In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction. We answer this question in the negative by constructing two infinitely smooth convex bodies of revolution about the $x_n$-axis in ${\mathbb R^n}$, $n \ge 3$, one origin symmetric and the other not centrally symmetric, with the same inner section function. Moreover, the pair of bodies can be arbitrarily close to the unit ball.

• "The geometry of p-convex intersection bodies," (with Jaegil Kim and Vladyslav Yaskin), Advances in Math., 226, (2011), no 6, 5320-5337.

Busemann's theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper we provide a version of Busemann's theorem for p-convex bodies. We show that the intersection body of a p-convex body is q-convex for certain q. Furthermore, we discuss the sharpness of the previous result by constructing an appropriate example. This example is also used to show that IK, the intersection body of K, can be much farther away from the Euclidean ball than K. Finally, we extend these theorems to some general measure spaces with log-concave and s-concave measures.

• "The behavior of iterations of the intersection body operator in a small neighborhood of the unit ball," (with Alexander Fish, Fedor Nazarov and Dmitry Ryabogin), Advances in Math., 226 (2011), no 3, 2629-2642.

The intersection body of a ball is again a ball. So, the unit ball $B_d$ in ${\mathbb R}^d$ is a fixed point of the intersection body operator acting on the space of all star-shaped origin symmetric bodies endowed with the Banach-Mazur distance. We show that this fixed point is a local attractor, i.e., that the iterations of the intersection body operator applied to any star-shaped origin symmetric body sufficiently close to B_d in Banach-Mazur distance converge to $B_d$ in Banach-Mazur distance. In particular, it follows that the intersection body operator has no other fixed or periodic points in a small neighborhood of $B_d$.

• "A remark on the Mahler conjecture: local minimality of the unit cube," (with Fedor Nazarov, Fedor Petrov and Dmitry Ryabogin), Duke Mathematical Journal, 154 (2010), no 3, 419-430.

We prove that the unit cube $B^n_\infty$ is a strict local minimizer for the Mahler volume product ${\rm vol}_n(K){\rm vol}_n(K^*)$ in the class of original symmetric convex bodies endowed with the Banach-Mazur distance. For more information check "Open question: the Mahler conjecture on convex bodies."

• "Gaussian Brunn-Minkowski-type inequlities," (with Richar J. Gardner). Trans. Amer. Math. Soc., 360, (2010), 10, 5333-5353.

A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski inequality is proved, together with precise equality conditions, and shown to be best possible from several points of view. A possible new Gaussian Brunn-Minkowski inequality is proposed, and proved to be true in some special cases. Throughout the study attention is paid to precise equality conditions and conditions on the coefficients of dilatation. Interesting links are found to the S-inequality and the (B) conjecture. An example is given to show that convexity is needed in the (B) conjecture.

• "On the local equatorial characterization of zonoids and intersection bodies," (with Fedor Nazarov and Dmitry Ryabogin), Advances in Math., 217 (2008), no. 3, 1368-1380.

In this paper we show that there is no local equatorial characterization of zonoids in odd dimensions. This gives a negative answer to the conjecture posed by W. Weil in 1977 and shows that the local equatorial characterization of zonoids may be given only in even dimensions. In addition we prove a similar result for intersection bodies and show that there is no local characterization of these bodies.

• "Gaussian measure of sections of dilates and shifts of convex bodies," Advances in Applied Mathematics, 41/2 (2008), 247-254.

In this paper we give a solution for the Gaussian version of the Busemann-Petty problem with additional information about dilates and shifts. We also provide some remark on the size of the Gaussian measure of the dilations of the unit cube.

• "The Busemann-Petty problem for arbitrary measures," Math. Ann. 331, (2005), 867-887.

The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections. We apply this formula to study properties of general measures most of which were known before only in the case of the standard Lebesgue measure. We solve an analog of the Busemann-Petty problem for the case of general measures. In addition, we show that there are measures, for which the answer to the generalized Busemann-Petty problem is affirmative in all dimensions. Finally, we apply the latter fact to prove a number of different inequalities concerning the volume of sections of convex symmetric bodies in R^n and solve a version of generalized Busemann-Petty problem for sections by k-dimensional subspaces.

• "An isomorphic version of the Busemann-Petty problem for Gaussian Measure," ".ps", GAFA Seminar Notes 2002-2003 Lecture Notes in Math, Vol. 1850, (2004), 277-283.

In this paper we provide upper and lower bounds for the Gaussian Measure of a hyperplane section of a convex body. We use those estimates to give a partial answer to an isomorphic version of the Gaussian Busemann-Petty problem.

• "Gaussian Measure of Sections of convex bodies," Advances in Math. 188/1 (2004), 124-136.

In this paper we study properties of sections of convex bodies with respect to the Gaussian measure. We develop a formula connecting the Minkowski functional of a convex symmetric body K with a Gaussian measure of its sections. Using this formula we solve an analog of the Busemann-Petty problem for Gaussian measures.

• "Reconstruction of convex bodies of revolution from the areas of their shadows," (with Dmitry Ryabogin), Archiv der Mathematik 5 (2004), 450-460.

In this short note we reconstruct a convex body of revolution from the areas of its shadows by giving a precise formula for the support function.

• "Fourier transform and Firey projections of convex bodies," (with Dmitry Ryabogin ), Indiana Univ. Math. J. 53 (2004), 667-682.

In this paper we develop a Fourier analytic approach to problems in the Brunn-Minkowski-Firey theory of convex bodies. We study the notion of Firey projections and prove a version of Aleksandrov's Theorem. We also formulate and solve an analog of the Shephard problem for Firey projections.

• "Projections of convex bodies and the Fourier transform," (with Alexander Koldobsky and Dmitry Ryabogin), Israel J. Math. 139 (2004), 361-380.

The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections of $\ell_p$-balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of the recent result of Barthe and Naor on extremal projections of $\ell_p$-balls, and give a Fourier analytic solution to Shephard's problem, originally solved by Petty and Schneider and asking whether symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function.

• "A remark on p-summing norms of operator," Trends in Banach spaces and Operator Theory, Contemporary Mathematics, 321 (Editor A. Kaminska), (2003), 371-378.

In this paper we improve a result of W. B. Johnson and G. Schechtman by proving that the p-summing norm of any operator with $n$-dimensional domain can be well-approximated using $C(p)n \log n(\log\log n)^2$ vectors if $1 \le p < 2$, and using $C(p)n^{p/2}\log n$ if $2 < p$.

• "Supremum of process in terms of trees", (with Olivier Guedon), GAFA Seminar Notes, 2001-2002, Lecture Notes in Math, Vol. 1807, (2003), 136-148.

In this paper we study the quantity ${\mathbb E} \sup X_t$, where $X_t$ is some random process. In the case of Gaussian process, there is a natural sub-metric d defined on T. We find an upper bound in terms of covering trees of $(T,d)$ and a lower bound in terms of packing trees (The idea to use these objects comes from works of M. Talagrand on majorizing measures). The two quantities are proved to be equivalent via a general result concerning packing trees and labelled-covering trees of a metric space. Instead of using majorizing measure theory, all the results involve the language of entropy numbers. Part of the results can be extended to some more general processes which satisfy some concentration inequality.

• "Isomorphic embedding of $\ell^n_p$, $1< p <2$, into $\ell^{(1+\varepsilon)n}_1$," (with Assaf Naor), Israel J. Math. 122 (2001), 371-380.

In this paper we partially answer a question posed by V. Milman and G. Schechtman by proving that for every $\varepsilon >0$ and $1< p< 2$, $\ell_p^n$, $K(n,p,\varepsilon)$-embeds into $\ell_1^{1+\varepsilon n}$, where $K(n,p,\varepsilon)$ is a power of $\log n$ depending only on $n$ and $\varepsilon$.

• "More on embedding subspaces of $L_p$ into $\ell^N_p$, $p\in (0,1)$," GAFA Seminar Notes 1996-2000, Lecture Notes in Math, Vol. 1745, pp. 269-280.

It is shown that if $X$ is $n$-dimensional subspace of $L_p$, $0< p<1$, then there exists a subspace $Y$ of $\ell_p^N$ such that $d_{BM}(X,Y)< 1+\varepsilon$ and $N< C(\varepsilon,p) n(\log n)( \log \log n)^2$.

• "Embedding subspaces of $L_p$ into $\ell^N_p$, $0< p <1$," (with Gideon Schechtman), Math. Nachr. 227 (2001), 133--142.

Given an $n$-dimensional subspace $X$ of $L_p$ and $\varepsilon >0$ what is the smallest integer $N=N(n,\varepsilon)$ such that there is a subspace $Y$ of $\ell_p^N$ with $d(X,Y)< 1+\varepsilon$, where $d(X,Y)$ is the Banach-Mazur distance? For $p\ge 1$ the dependence of $N$ on $n$ is known, exept for possibly redundant $\log$ factors. It is $n \log n$ for $p=1$ and $n \log n( \log \log n)^2$ for $1< p <2$ ( M. Talagrand ) and $n(\log n)^3$ for $2< p$ (J. Bourgain, J. Lindenstrauss and V. Milman ). In this paper we extend these results to the range $0< p < 1$ by proving that $X$ is $n$-dimensional subspace of $L_p$, $0< p< 1$, then there exists a subspace $Y$ of $\ell_p^N$ such that $d(X,Y) < 1+\varepsilon$ and $N < C(\varepsilon,p) n(\log n)^3$.

### Ph.d. thesis (2001)

"Finite dimensional subspaces of $L_p$, $0 < p < \infty$" (advisor: Gideon Schechtman), ".dvi".

### M.Sc. thesis (1996)

"The critical probability for Voronoi percolation", (advisor: Oded Schramm), ".dvi (zip)".