Publications

(the files below might be slightly different from the published versions) 

2024 and preprints

• Horocyclic Brunn-Minkowski inequality - joint with R. Assouline. Adv. Math., Vol. 436:109381, (2024), 39 pp.

• A weak version of the epsilon-Dvoretzky conjecture for normed spaces - joint with T. Novikov. To appear in Israel Journal of Math.

• The strong data processing inequality under the heat flow - joint with O. Ordentlich. Preprint.

• Isoperimetric inequalities in high-dimensional convex sets - joint with J. Lehec. To appear in Bull. Amer. Math. Soc. 

• Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound - joint with J. Lehec. Preprint. Here are related slides and video.

2023

• Local tail bounds for polynomials on the discrete cube - joint with S. Sodin. Geometric Aspects of Functional Analysis, Lecture Notes in Math. 2327, Springer (2023), 223--230.

• Long lines in subsets of large measure in high dimension - joint with D. Elboim. Probab. Theory Related Fields 187, no. 3-4, (2023), 657–695. Here are related slides and video.

• Linear extension operators for Sobolev spaces on uniform trees - joint with C. Fefferman. Advanced Nonlinear Studies, vol. 23: 20220075, no. 1, (2023), 18 pp.

• Logarithmic bounds for isoperimetry and slices of convex sets. Ars Inveniendi Analytica (2023), Paper No. 4, 17 pp.

• Spectral monotonicity under Gaussian convolution - joint with E. Putterman. Ann. Fac. Sci. Toulouse Math, Serie 6, Vol. 32, no. 5,  (2023), 939--967.

2022

• The slicing problem by Bourgain - joint with V. Milman. Analysis at Large, Dedicated to the
  Life and Work of Jean Bourgain, edited by A. Avila, M. Rassias and Y. Sinai, Springer, (2022), 203–232.
   
• Bourgain's slicing problem and KLS isoperimetry up to polylog - joint with J. Lehec.  Geometric and Functional Analysis (GAFA), Vol. 32, Springer (2022), 1134–1159. 

2021 

• Rigidity of Riemannian embeddings of discrete metric spaces - joint with M. Eilat. Invent.
  Math., Vol. 226, (2021), 349—391. Here is a relevant video and a notebook.

• Discrete variants of Brunn-Minkowski type inequalities - joint with D. Halikias and Boaz Slomka. Ann. Fac. Sci. Toulouse Math. (6), Vol. 30, no. 2, (2021), 267—279.

2020 

• Complex Legendre duality - joint with B. Berndtsson, D. Cordero-Erausquin and Y. A. Rubinstein. American Journal of Math., Vol. 142, No. 1, (2020). Here are related slides.
• Complex interpolation of R-norms, duality and foliations - joint with B. Berndtsson, D. Cordero-Erausquin and Y. A. Rubinstein. J. Euro. Math. Soc., Vol. 22, Issue 2, (2020), 477–505. Here are related slides.
• A coordinate-free proof of the finiteness principle for the Whitney extension problem - joint with J. Carruth, A. Frei-Pearson and A. Israel. Rev. Mat. Iberoam., Vol. 36, no. 7, (2020), 1917-–1956. Here is a clarification of a misleading remark from this paper.
• Unimodal value distribution of Laplace eigenfunctions and a monotonicity formula. Geom. Dedicata, Vol. 208, (2020), 13–-29.
• Chasing nested convex bodies nearly optimally - joint with S. Bubeck, Y.-T. Lee, Y. Li, M. Sellke. Symposium on Discrete Algorithms (SODA 2020), 1496–-1508. Here is a relevant video and a notebook.
• The lower bound for Koldobsky's slicing inequality via random rounding - joint with G. V. Livshyts. Geometric Aspects of Functional Analysis, Lecture Notes in Math. 2266, Springer (2020), 43--63. [Erratum: Julian Haddad noted an inaccuracy in Section 4 of the published version. We fixed this issue, and the corrected paper is available here and on arXiv].
• Spectral gaps, symmetries and log-concave perturbations - joint with F. Barthe. Bull. Hellenic Math. Soc., Vol. 64 (2020), 1–-31.

2019 

• Poisson processes and a log-concave Bernstein theorem - joint with J. Lehec. Studia Math., Vol. 247, (2019), 85–107.
• Extremal Kahler-Einstein metric for two-dimensional convex bodies - joint with A. V. Kolesnikov. Journal of Geometric Analysis, Vol. 29, Issue 3, (2019), 2347–2373.
• One more proof of the Alexandrov-Fenchel inequality - joint with D. Cordero-Erausquin, Q. Merigot, F. Santambrogio. Comptes Rendus Mathematique, Volume 357, Issue 8, (2019), 676--680. Here is a related video and a notebook. 

2018

• Isotropic constants and Mahler volumes. Advances in Math., Vol. 330, (2018), 74–108. Here are lecture notes from the Oberwolfach reports.
• An example related to the slicing inequality for general measures - joint with A. Koldobsky. J. Funct. Analysis, Vol. 274, Issue 7, (2018), 2089–2112.
• Estimates for moments of general measures on convex bodies - joint with S. Bobkov and A. Koldobsky. Proc. Amer. Math. Soc., Vol. 146, No. 11, (2018), 4879–4888.
• Eldan's stochastic localization and tubular neighborhoods of complex-analytic sets. Journal of Geometric Analysis, Vol. 28, Issue 3, (2018), 2008–2027.

2017 

• Optimal compression of approximate inner products and dimension reduction - joint with N. Alon. Symposium on Foundations of Computer Science (FOCS 2017), 639–650. Here are related slides and video.
• Needle decompositions in Riemannian geometry. Mem. Amer. Math. Soc., Vol. 249, No. 1180, (2017). Here are related slides and video.
• Remarks on curvature in the transportation metric - joint with A. V. Kolesnikov. Analysis Math., Vol. 43, No. 1, (2017), 67–88.
• Convex geometry and waist inequalities. Geom. Funct. Anal. (GAFA), Vol. 27, No. 1, (2017), 130–164. Here is a related video.
• Super-Gaussian directions of random vectors. Geometric Aspects of Functional Analysis, Lecture Notes in Math. 2169, Springer (2017), 187--211. Here are related slides.
• Affine hemispheres of elliptic type. Algebra i Analiz (St. Petersburg Math. Journal), Vol. 29, Issue 1, (2017), 145–188. Here is an STL file for 3D-printing an affine hemisphere with a square base, courtesy of Quentin Merigot and Filippo Santambrogio.

2016 

• Discrete curvature and abelian groups - joint with G. Kozma, P. Ralli and P. Tetali. Canad. J. Math. Vol. 68, No. 3, (2016), 655–674.
• Lipschitz functions on the infinite-dimensional torus - joint with D. Faifman. Commun. Contemp. Math., Vol. 18, No. 1 (2016), 1550029

2015

• Moment measures - joint with D. Cordero-Erausquin. J. Functional Analysis, Vol. 268, No. 12, (2015), 3834-–3866. Here are related slides.
• Eigenvalue distribution of optimal transportation - joint with A. V. Kolesnikov. Analysis & PDE., Vol. 8, No. 1, (2015), 33–-55. Here is a related video.

2014

• Logarithmically-concave moment measures I. Geometric Aspects of Functional Analysis, Lecture Notes in Math. 2116, Springer (2014), 231--260.
• On the oscillation rigidity of a Lipschitz function on a high-dimensional flat torus - joint with D. Faifman and V. D. Milman. Geometric Aspects of Functional Analysis, Lecture Notes in Math. 2116, Springer (2014), 123--131.
• Concentration of measures supported on the cube. Israel J. of Math., Vol. 203, Issue 1, (2014), 59--80.
• Dimensionality and the stability of the Brunn-Minkowski inequality - joint with R. Eldan. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Vol. XIII, (2014), 975--1007.

2013

• Poincare inequalities and moment maps. Ann. Fac. Sci. Toulouse Math., Vol. 22, No. 1, (2013), 1--41. Here are related slides.

2012

• Inner regularization of log-concave measures and small-ball estimates - joint with E. Milman. Geometric Aspects of Functional Analysis, Lecture Notes in Math. 2050, Springer (2012), 267--278.
• Interpolations, convexity and geometric inequalities - joint with D. Cordero-Erausquin. Geometric Aspects of Functional Analysis, Lecture Notes in Math. 2050, Springer (2012), 151--168.
• Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality - joint with S. Artstein-Avidan, C. Schütt and E. Werner. J. Functional Analysis, Vol. 262, No. 9, (2012), 4181--4204.
• Centroid bodies and the logarithmic Laplace transform - a unified approach - joint with E. Milman. J. Functional Analysis, Vol. 262, No. 1, (2012), 10--34. Here are related slides.

2011

• Variations on the Berry-Esseen theorem - joint with S. Sodin. Theory Probab. Appl. (also known as "Teoriya Veroyatnostei i ee Primeneniya"), Vol. 56, No. 3, (2011), 514--533.
• Approximately gaussian marginals and the hyperplane conjecture - joint with R. Eldan. Proc. of a workshop on "Concentration, Functional Inequalities and Isoperimetry", Contermporary Mathematics 545, Amer. Math. Soc., (2011), 55--68. Here are related slides.
• Quantum One-Way Communication is Exponentially Stronger Than Classical Communication - joint with O. Regev. STOC 2011 (Proceedings of a Symposium on Theory of Computing), Assoc. Comput. Mach., (2011), 31--40. Here are related slides.

2010

• On nearly radial marginals of high-dimensional probability measures. J. Eur. Math. Soc, Vol. 12, (2010), 723–-754. Here are related slides.
• High-dimensional distributions with convexity properties. Proc. of the Fifth Euro. Congress of Math., Amsterdam, July 2008. Eur. Math. Soc. publishing house, (2010), 401--417. Here are related slides and a loosely related video.

2009

• An example related to Whitney extension with almost minimal C^m norm - joint with C. Fefferman, Rev. Mat. Iberoamericana, 25, no. 2, (2009), 423--446.
• Economical toric spines via Cheeger's Inequality - joint with N. Alon. Journal of Topology and Analysis, vol. 1, (2009), 101--111.
• A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields, vol. 45, no. 1, (2009), 1 -- 33.
• On the hyperplane conjecture for random convex sets - joint with G. Kozma. Israel journal of math., vol. 170, no. 1, (2009), 253--268.
• Fitting a C^m-Smooth Function to Data I - joint with C. Fefferman, Ann. of Math., 169, no. 1, (2009), 315--346.
• Fitting a C^m-Smooth Function to Data II - joint with C. Fefferman, Rev. Mat. Iberoamericana, 25, no. 1, (2009), 49--273.

2008

• Pointwise Estimates for Marginals of Convex Bodies - joint with R. Eldan, J. Functional Analysis, Vol. 254, Issue 8, (2008), 2275--2293.
• On volume distribution in 2-convex bodies - joint with E. Milman. Israel J. Math., Vol. 164 (2008), 221--249.

2007

• Uniform almost sub-gaussian estimates for linear functionals on convex sets, Algebra i Analiz (St. Petersburg Math. Journal), Vol. 19, no. 1 (2007), 109--148.
• A central limit theorem for convex sets, Invent. Math., Vol. 168, (2007), 91--131. (A neat proof of Lemma 4.9 using Pinsker's inequality can be found here).
• Power-law estimates for the central limit theorem for convex sets, J. Functional Analysis, Vol. 245, (2007), 284--310.
• Marginals of geometric inequalities, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1910, Springer (2007), 133-166.
• C^1 extensions of functions and stabilization of Glaeser refinements - joint with N. Zobin. Rev. Mat. Iberoamericana, Vol. 23, no. 2 (2007), 635--669.
• Small ball probability and Dvoretzky Theorem - joint with R. Vershynin, Israel J. Math., Vol. 157, no. 1 (2007), 193--207.

2006

• Isomorphic and almost-isometric problems in high-dimensional convex geometry, proceedings of the Internat. Congress of Mathematicians, Madrid, Spain, 2006. Eurpoean Math. Soc., Vol. II, (2006), 1547--1562.
• On convex perturbations with a bounded isotropic constant, Geometric and Functional Analysis (GAFA), Vol. 16, Issue 6 (2006) 1274--1290.

2005

• Geometry of log-concave functions and measures - joint with V. Milman, Geom. Dedicata 112 (2005) 169--182.
• An isomorphic version of the slicing problem, J. Functional Analysis, Vol. 218 (2005) 372 -- 394.
• Rapid Steiner symmetrization of most of a convex body and the slicing problem - joint with V. Milman. Combin. Probab. Comput. 14, no. 5-6 (2005) 829--843. [Erratum: The proof of Proposition 2.2 is bogus, fortunately a fix was found by Vritsiou]. 
• Empirical processes and random projections - joint with S. Mendelson, J. Functional Analysis, Vol. 225, no. 1 (2005) 229--245.

2004

• The Santalo point of a function, and a functional form of Santalo inequality - joint with S. Artstein, V. Milman, Mathematika 51 (2004) 33--48.
• Rate of convergence of geometric symmetrization, Geometric and Functional Analysis (GAFA), Vol 14, Issue 6 (2004) 1322--1338.
• Symmetrization and isotropic constants of convex bodies - joint with J. Bourgain, V. Milman. Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1850, Springer (2004) 101-116.
• On John-type ellipsoids, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1850, Springer (2004) 149-158.
• A geometric inequality and a low M-estimate, Proc. Amer. Math. Soc., Vol. 132, No. 9, 2919-2628 (2004)

2003

• A reduction of the slicing problem to finite volume ratio bodies - joint with J. Bourgain, V. Milman. C. R. Math. Acad. Sci. Paris 336 (2003), no. 4, 331-334.
• Isomorphic Steiner symmetrization - joint with V. Milman. Invent. Math. 153 (2003), no. 3, 463--485.

2002

• 5n Minkowski symmetrizations suffice to arrive at an approximate Euclidean ball, Ann. of Math. 156 (2002), no. 3, 947--960.

2000

• Remarks on Minkowski symmetrizations, Geometric Aspects of Functional Analysis, Lecture Notes in Math. 1745, Springer (2000) 109--117.

Ph.D. Thesis:

My PhD thesis (and its short Hebrew version), written under the supervision of Prof. Vitali Milman at Tel-Aviv University.