Artem Zvavitch - Academia.edu (original) (raw)
Papers by Artem Zvavitch
International Mathematics Research Notices, 2016
Indiana University Mathematics Journal, 2004
In this paper we develop a Fourier analytic approach to problems in the Brunn-Minkowski-Firey the... more 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 projection theorem. We also formulate and solve an analog of the Shephard problem for Firey projections.
In addition, we deduce the 1/n1/n1/n-concavity of the parallel volume tmapstomu(A+tB)t \mapsto \mu(A+tB)tmapstomu(A+tB), Brunn's ... more In addition, we deduce the 1/n1/n1/n-concavity of the parallel volume tmapstomu(A+tB)t \mapsto \mu(A+tB)tmapstomu(A+tB), Brunn's type theorem and certain analogues of Minkowski first inequality.
A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. ... more A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski inequality, the first of its type, is proved, together with precise equality conditions, and is shown to be the best possible from several points of view. A new Gaussian Brunn-Minkowski inequality is proposed and proved to be true in some significant 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.
In this paper we study the quantity E sup t∈T Xt, where Xt is some random process. In the case of... more In this paper we study the quantity E sup t∈T Xt, where Xt is some random process. In the case of the Gaussian process, there is a natural sub-metric d defined on T . We find an upper bound in terms of labelled-covering trees of (T, d) and a lower bound in terms of packing trees (this uses the knowledge of packing numbers of subsets of T ). 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 the 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.
The intersection body of a ball is again a ball. So, the unit ball BdsubsetRdB_d \subset \R^dBdsubsetRd is a fixed ... more The intersection body of a ball is again a ball. So, the unit ball BdsubsetRdB_d \subset \R^dBdsubsetRd 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
Comptes Rendus Mathematique, 2016
Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its... more 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 R^n. The corresponding inequality to the conjecture is sometimes called the the reverse Blaschke-Santalo inequality. The conjecture is known in dimension two 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.
Duke Mathematical Journal, Sep 15, 2010
We prove that the unit cube B n ∞ is a strict local minimizer for the Mahler volume product vol n... more We prove that the unit cube B n ∞ is a strict local minimizer for the Mahler volume product vol n (K)vol n (K * ) in the class of origin symmetric convex bodies endowed with the Banach-Mazur distance.
Journal of Differential Geometry, Jan 17, 2011
In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and refle... more 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 xnx_nxn-axis in Rn\R^nRn, nge3n\ge 3nge3, 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.
Springer Proceedings in Mathematics & Statistics, 2012
ABSTRACT We discuss some open questions on unique determination of convex bod-ies.
Fourier Analysis and Convexity, 2004
It has been noticed long ago that many results on sections and projections are dual to each other... more 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. In the paper [KRZ], the authors attempted to start a unified approach connecting sections and projections, which may eventually explain these mysterious connections. The idea is to use the recently developed Fourier analytic approach to sections of convex bodies (a short description of this approach can be found in [K7]) as a prototype of a new approach to projections. The first results seem to be quite promising. The crucial role in the Fourier approach to sections belongs to certain formulas connecting the volume of sections with the Fourier transform of powers of the Minkowski functional. An analog of these formula for the case of projections was found in [KRZ] and connects the volume of projections to the Fourier transform of the curvature function. This formula was applied in [KRZ] to give a new proof of the result of Barthe and Naor on the extremal projections of l p -balls with p > 2, which is similar to the proof of the result on the extremal sections of l p -balls with 0 < p < 2 in [K5]. Another application is to the Shephard problem, asking whether bodies with smaller hyperplane projections necessarily have smaller volume. The problem was solved independently by Petty and Schneider, and the answer is affirmative in the dimension two and negative in the dimensions three and higher. The paper [KRZ] gives a new Fourier analytic solution to this problem that essentially follows the Fourier analytic solution to the Busemann-Petty problem (the projection counterpart of Shephard's problem) from [K3]. The transition in the Busemann-Petty problem occurs between the dimensions four and five. In Section 4, we show that the transition in both problems has the same explanation based on similar Fourier analytic characterizations of intersection and projection bodies.
Lecture Notes in Mathematics, 2003
In this paper we study the quantity E sup t∈T Xt, where Xt is some random process. In the case of... more In this paper we study the quantity E sup t∈T Xt, where Xt is some random process. In the case of the Gaussian process, there is a natural sub-metric d defined on T . We find an upper bound in terms of labelled-covering trees of (T, d) and a lower bound in terms of packing trees (this uses the knowledge of packing numbers of subsets of T ). 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 the 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.
Geometriae Dedicata, 2014
In this paper we provide upper and lower bounds for the Gaussian Measure of a hyperplane section ... more In this paper we provide upper and lower bounds for the Gaussian Measure of a hyperplane section of a convex symmetric body. We use those estimates to give a partial answer to an isomorphic version of the Gaussian Busemann-Petty problem.
International Mathematics Research Notices, 2016
Indiana University Mathematics Journal, 2004
In this paper we develop a Fourier analytic approach to problems in the Brunn-Minkowski-Firey the... more 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 projection theorem. We also formulate and solve an analog of the Shephard problem for Firey projections.
In addition, we deduce the 1/n1/n1/n-concavity of the parallel volume tmapstomu(A+tB)t \mapsto \mu(A+tB)tmapstomu(A+tB), Brunn's ... more In addition, we deduce the 1/n1/n1/n-concavity of the parallel volume tmapstomu(A+tB)t \mapsto \mu(A+tB)tmapstomu(A+tB), Brunn's type theorem and certain analogues of Minkowski first inequality.
A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. ... more A detailed investigation is undertaken into Brunn-Minkowski-type inequalities for Gauss measure. A Gaussian dual Brunn-Minkowski inequality, the first of its type, is proved, together with precise equality conditions, and is shown to be the best possible from several points of view. A new Gaussian Brunn-Minkowski inequality is proposed and proved to be true in some significant 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.
In this paper we study the quantity E sup t∈T Xt, where Xt is some random process. In the case of... more In this paper we study the quantity E sup t∈T Xt, where Xt is some random process. In the case of the Gaussian process, there is a natural sub-metric d defined on T . We find an upper bound in terms of labelled-covering trees of (T, d) and a lower bound in terms of packing trees (this uses the knowledge of packing numbers of subsets of T ). 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 the 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.
The intersection body of a ball is again a ball. So, the unit ball BdsubsetRdB_d \subset \R^dBdsubsetRd is a fixed ... more The intersection body of a ball is again a ball. So, the unit ball BdsubsetRdB_d \subset \R^dBdsubsetRd 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
Comptes Rendus Mathematique, 2016
Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its... more 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 R^n. The corresponding inequality to the conjecture is sometimes called the the reverse Blaschke-Santalo inequality. The conjecture is known in dimension two 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.
Duke Mathematical Journal, Sep 15, 2010
We prove that the unit cube B n ∞ is a strict local minimizer for the Mahler volume product vol n... more We prove that the unit cube B n ∞ is a strict local minimizer for the Mahler volume product vol n (K)vol n (K * ) in the class of origin symmetric convex bodies endowed with the Banach-Mazur distance.
Journal of Differential Geometry, Jan 17, 2011
In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and refle... more 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 xnx_nxn-axis in Rn\R^nRn, nge3n\ge 3nge3, 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.
Springer Proceedings in Mathematics & Statistics, 2012
ABSTRACT We discuss some open questions on unique determination of convex bod-ies.
Fourier Analysis and Convexity, 2004
It has been noticed long ago that many results on sections and projections are dual to each other... more 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. In the paper [KRZ], the authors attempted to start a unified approach connecting sections and projections, which may eventually explain these mysterious connections. The idea is to use the recently developed Fourier analytic approach to sections of convex bodies (a short description of this approach can be found in [K7]) as a prototype of a new approach to projections. The first results seem to be quite promising. The crucial role in the Fourier approach to sections belongs to certain formulas connecting the volume of sections with the Fourier transform of powers of the Minkowski functional. An analog of these formula for the case of projections was found in [KRZ] and connects the volume of projections to the Fourier transform of the curvature function. This formula was applied in [KRZ] to give a new proof of the result of Barthe and Naor on the extremal projections of l p -balls with p > 2, which is similar to the proof of the result on the extremal sections of l p -balls with 0 < p < 2 in [K5]. Another application is to the Shephard problem, asking whether bodies with smaller hyperplane projections necessarily have smaller volume. The problem was solved independently by Petty and Schneider, and the answer is affirmative in the dimension two and negative in the dimensions three and higher. The paper [KRZ] gives a new Fourier analytic solution to this problem that essentially follows the Fourier analytic solution to the Busemann-Petty problem (the projection counterpart of Shephard's problem) from [K3]. The transition in the Busemann-Petty problem occurs between the dimensions four and five. In Section 4, we show that the transition in both problems has the same explanation based on similar Fourier analytic characterizations of intersection and projection bodies.
Lecture Notes in Mathematics, 2003
In this paper we study the quantity E sup t∈T Xt, where Xt is some random process. In the case of... more In this paper we study the quantity E sup t∈T Xt, where Xt is some random process. In the case of the Gaussian process, there is a natural sub-metric d defined on T . We find an upper bound in terms of labelled-covering trees of (T, d) and a lower bound in terms of packing trees (this uses the knowledge of packing numbers of subsets of T ). 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 the 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.
Geometriae Dedicata, 2014
In this paper we provide upper and lower bounds for the Gaussian Measure of a hyperplane section ... more In this paper we provide upper and lower bounds for the Gaussian Measure of a hyperplane section of a convex symmetric body. We use those estimates to give a partial answer to an isomorphic version of the Gaussian Busemann-Petty problem.