Convex Geometry
Geometric Functional Analysis
Probability
Harmonic Analysis
Gaussian Brunn-Minkowski-type inequlities.(with Richar J. Gardner). Trans. Amer. Math. Soc., accepted.
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" to appear in Advances in Applied Mathematics.
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.
"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.
"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 l_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 l_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(loglog n)^2 vectors if 1< p< 2, and using C(p)n^(p/2)log n if 2 < p.
"Supremum of process in terms of trees", (with Olivier Guﾁédon ), GAFA Seminar Notes, 2001-2002, Lecture Notes in Math, Vol. 1807, (2003), 136-148.
In this paper we study the quantity 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 l^n_p , 1< p < 2 , into l^{(1+\e)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 0< e and 1< p< 2, l_p^n, K(n,p,e)-embeds into l_1^(en), where K(n,p,e) is a power of log n depending only on n and e.
"More on embedding subspaces of L_p into l^N_p , 0< p < 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 l_p^N such that d(X,Y) < 1+e and N < C(e,p) n(log n)( log log n)^2.
"Embedding subspaces of L_p into l^N_p , 0< p < 1 ", (with Gideon Schechtman), Math. Nachr. 227 (2001), 133--142.
Given an n-dimensional subspace X of L_p and 0< e what is the smallest integer N=N(n,e) such that there is a subspace Y of l_p^N with d(X,Y)< 1+e, 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 l_p^N such that d(X,Y) < 1+e and N < C(e,p) n(log n)^3.
"Finite dimensional subspaces of Lp, 0 < p < \infty" (advisor: Gideon Schechtman), ".dvi".
"The critical probability for Voronoi percolation", (advisor: Oded Schramm), ".dvi (zip)" version.