- Curriculum Vitae
- Teaching
- Research Topics
- Convex Geometry
- Analysis
- Probability
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Applications of the above to Approximation Theory, Mathematical Physics, and Quantum
Information Theory
- Conference Organisation
- Interplay of convex geometry and Banach space theory, Banff, Canada, March 2013.
- Invariants in convex geometry and Banach space theory, AIM, Palo Alto, August 2012.
- Geometry of Quantum Entanglement, CIRM, Marseille, January 2012.
- Perspectives in High Dimensions, Cleveland, Ohio, August 2010.
- Geometry of Quantum Entanglement, Oberwolfach, Germany, December 2009.
- Affine Convex Geometric Analysis, Banff, Canada, January 2009.
- Fourier Analytic and Probabilistic Methods in Functional Analysis and Convexity, Kent, Ohio, August 2008.
- Special Session on Convex Geometry, Hoboken, New Jersey, April 2007.
- Asymptotic Analysis and Applications, Paris, France, August, 2006.
- Convex Geometric Analysis, Banff, Canada, July 2004.
- Special Session "Analytic Convex Geometry", Lawrenceville, New Jersey, April 2004.
- Special Session in "Convex Geometry", Boston, Massachusetts, October 2002.
- Special Session on "Invariance Convex Geometry", Lowell, Massachusetts, April 2000.
- Special Session on "Harmonic Analysis and Convexity", Memphis, Tennissee, March 1997.
- Publications
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On the mixed $f$-divergence for multiple pairs of measures
Jointly with Deping Ye.
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Affine invariant points
Jointly with M. Meyer and C. Schuett.
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f-Divergence for convex bodies
Proceedings of the ``Asymptotic Geometric Analysis" workshop, Fields Institute, Toronto 2012
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On the approximation of a polytope by its dual Lp-centroid bodies
to appear in Indiana Univ. Math. J.
Jointly with G. Paouris.
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Relative entropies for convex bodies
to appear in TRANSACTIONS OF THE AMS
Jointly with J. Jenkinson.
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Renyi Divergence and Lp-affine surface area for convex bodies
Advances in Mathematics 230, 1040--1059 (2012)
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Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality
Journal of Functional Analysis 262, 4181-4204 (2012)
Jointly with S. Artstein-Avidan, B. Klartag and C. Schuett
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Relative entropy of cone measures and Lp centroid bodies
PROCEEDINGS LONDON MATH. SOC. 104 (2012), no. 2, 253–286.
Jointly with G. Paouris.
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A note on Mahler's conjecture
INTERNATIONAL MATH. RESEARCH NOTICES, (2012), no. 1, 1–16.
Jointly with S. Reisner and C. Schuett.
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Uniform estimates for order statistics and Orlicz functions
POSITIVITY, 16 (2012), no. 1, 1–28.
Jointly with Y. Gordon, A. Litvak and C. Schuett.
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On the Homothety Conjecture
Indiana Univ. Math. J. 60 No. 1, 1–20, (2011)
Jointly with Deping Ye.
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New affine measures of symmetry for convex bodies
ADVANCES IN MATHEMATICS 228, 2920–2942 (2011)
Jointly with M. Meyer and C. Schuett.
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How often is a random quantum state k-entangled
J. PHYSICS A: MATH. THEOR. 44 (2011)
Jointly with S. Szarek and K. Zyczkowski.
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Hastings' additivity counterexample via Dvoretzky's theorem
COMMUNICATIONS IN MATHEMATICAL PHYSICS 305 (2011)
Jointly with G. Aubrun and S. Szarek.
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Non-additivity of Renyi entropy and Dvoretzky's Theorem
JOURNAL OF MATHEMATICAL PHYSICS 51 (2010) 022102 and
VIRTUAL JOURNAL OF QUANTUM INFORMATION 10 (2010)
Jointly with G. Aubrun and S. Szarek.
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Inequalities for mixed p-affine surface area
MATHEMATISCHE ANNALEN 347, 703-737 (2010)
Jointly with Deping Ye.
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New Higher Order Equiaffine Invariants
ISRAEL JOURNAL OF MATHEMATICS 171, 221-235 (2009)
Jointly with A. Stancu.
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Bipartite states of low rank are almost surely entangled
J. PHYS. A: MATH. THEOR. 42 (2009) 095303
Jointly with M. B. Ruskai
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New Lp affine isoperimetric inequalities
ADVANCES IN MATHEMATICS 218, 762-780 (2008)
Jointly with Deping Ye
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Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
J. MATH. PHYS. 49, no. 3, 032113 (2008)
Jointly with S. Szarek and K. Zyczkowski
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On Lp affine surface areas
INDIANA UNIVERSITY MATH. J. 56, 2305-2324 (2007)
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On the minimum of several random variables
PROCEEDINGS AMERICAN MATHEMATICAL SOCIETY 134, 3665-3675 (2006)
Jointly with Y. Gordon, A. Litvak and C. Schuett
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Floating bodies and Illumination bodies
PROCEEDINGS OF THE CONFERENCE "INTEGRAL GEOMETRY AND CONVEXITY"
Wuhan 2004, World Scientific, Singapore (2006)
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Approximation of the Euclidean ball by a polytope
STUDIA MATHEMATICA 173, 1-18 (2006)
Jointly with M. Ludwig and C. Schuett
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Minima of sequences of Gaussian random variables
COMPTES RENDUS DE L'ACADEMIE DES SCIENCES PARIS, Ser.I 340, 445-448 (2005)
Jointly with Y. Gordon, A. Litvak and C. Schuett
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Surface bodies and p - affine surface area
ADVANCES IN MATHEMATICS 187, 98-145 (2004)
Jointly with C. Schuett.
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Geometry of spaces between polytopes and related zonotops
BULLETIN DES SCIENCES MATHEMATIQUES 126, 733-762 (2002)
Jointly with Y. Gordon, A. Litvak and C. Schuett
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Orlicz norms of sums of random variables
ANNALS OF PROBABILITY 30 , 1833-1853 (2002)
Jointly with Y. Gordon, A. Litvak and C. Schuett
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The p - affine surface area and geometric interpretations
RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO (2) Suppl. 70, part II, 367-382 (2002)
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Random polytopes of points chosen from the boundary of a convex body
GAFA SEMINAR NOTES, LECTURE NOTES IN MATHEMATICS 1807, Springer, 241-422 (2002)
Jointly with C. Schuett
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An Analysis of Completely positive Trace preserving maps on M2
LINEAR ALGEBRA AND ITS APPLICATIONS 347, 159-187 (2002)
Jointly with M. B. Ruskai and S. Szarek
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Dropping a vertex or a facet from a convex polytope
FORUM MATHEMATICUM 13, 359-378 (2001)
Jointly with S. Reisner and C. Schuett
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Random polytopes with vertices on the boundary of a convex body
COMPTES RENDUS DE L'ACADEMIE DES SCIENCES PARIS 331, 697-201 (2000)
Jointly with C. Schuett
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Study of a class of regularizations of 1/x using Gaussian integrals
SIAM J.OURNAL OF MATHEMATICAL ANALYSIS vol.32, no. 2, 435-463 (2000)
Jointly with M. B. Ruskai
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One Dimensional Regularizations of the Coulomb Potential with Application to Atoms in Strong Magnetic Fields
DIFFERENTIAL EQUATIONS AND MATHEMATICAL PhYSICS, 43-51 International Press, (2000)
Jointly with M. B. Ruskai and R. Brummelhuis
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On the p-affine surface area
ADVANCES IN MATHEMATICS 152, 288-313 (2000)
Jointly with M. Meyer
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A general geometric construction of affine surface area
STUDIA MATHEMATICA 132, 227-238 (1999)
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Confidence Regions for Means of Multivariate Normal Distributions and a non-symmetric Correlation Inequality for Gaussian measure
JOURNAL OF MULTIVARIATE ANALYSIS 68, 193-211 (1999)
Jointly with S. Szarek
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The Santalo-regions of a convex body
TRANSACTIONS OT THE AMS 350, 4569-4591 (1998)
Jointly with M. Meyer
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The illumination body of almost polygonal bodies
GEOMETRIAE DEDICATA 64, 343-354 (1997)
- Projects
- 2000 - 2015 NSF: Convexity and Applications
- 2007 - 2011 BSF: Applications of the methods of Asymptotic Geometric Analysis to Symplectic Geometry, Functional Inequalities and Metric Entropy
- 2007 - 2012 FRG Collaborative Research Grant: Fourier Analytic and Probabilistic Methods in Geometric Functional Analysis and Convexity
- 1994 - 1997 NSF: Convexity and Applications
- 1989 - 1993 NSF: Banach Spaces and Convexity
- Ph.D Students
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Since 2011 Umut Caglar
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2009 - 2013 Justin Jenkinson
Thesis: "Convex Geometric Connections to Information Theory"
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2006 - 2009 Deping Ye
Thesis: "Topics in Convex Geometry and Phenomena in High Dimension"
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2001 - 2003 Mark Meckes
Thesis: "Random Phenomena in Finite Dimensional Normed Spaces"