I will be at ICERM, Brown University, in the Fall 2022 semester
Curriculum Vitae
Teaching
Research Topics
Convex Geometry
Analysis
Probability
Numberphile
Simons Fellowship 2020-2021
Trimester Program, Hausdorff Institut
Conference Organisation
Workshop "High Dimensional Probability", Bedlewo, Poland, June 2023.
Special Session "Convex geometry and PDE"
CMS summer meeting, Memorial University, St. John's Canada
June 2022.
Workshop "Stochastic Aspects in Convexity", University of Bochum, Germany, May 2022.
Calculus of Variations in Probability and Geometry , IPAM, Los Angeles, February 2022.
Interaction between Partial Differential Equations and Convex Geometry ,
BIRS Hangzhu, China, October 2021
Special Session: Convexity and Probability in High Dimensions,
AMS online meeting University of Tennessee, Chattanooga
October 2020.
New Perspectives and Computational Challenges in High Dimensions, Oberwolfach, Germany February 2020.
Symmetry and Convexity in geometric inequalities , AIM, San Jose, May 2019.
Special Session: Convex Geometry and Functional Inequalities ,
University of Delaware, September 2018
Convex Geometry and Geometric Functional Analysis ,
TSIMF, Sanya, China, May 2018
Emerging Trends in Geometric Functional Analysis ,
Banff, Canada, March 2018
Special Session on "Probability in Convexity and Convexity in Probability" ,
OSU, Columbus, OH March 2018
Connections for Women: geometry and probability in high dimensions
,
MSRI, Berkeley, California, August 2017
"Perspectives in High-dimensional Probability and Convexity" , Oberwolfach, Germany, February 2017.
Special Session on "Discrete Structures: Analysis and Applications" , Minneapolis/St. Paul, October, 2016.
Special Session on "Probabilistic and Analytic tools in Convexity" , Athens, Georgia, March 2016.
Affine Geometric Analysis , BIRS Oaxaca, September 2015.
Asymptotic Geometric Analysis ,
Case Western Reserve University, August 2015.
Special Session on "
High dimensional convexity and applications" , San Francisco, October 25-26, 2014.
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, Tennessee, March 1997.
Publications
Curvature functionals on convex bodies, submitted.
Jointly with Kateryna Tatarko.
Affine Surface Area, submitted.
Jointly with Carsten Schuett.
Spherical convex hull of random points on a wedge,
submitted.
Jointly with Florian Besau, Anna Gusakova, Matthias Reitzner, Carsten Schuett, Christoph Thaele.
Ulam floating functions,
submitted.
Jointly with Chunyan Liu, Deping Ye, and Ning Zhang.
Geometric representation of classes of concave functions and duality
arXiv:2112.13881, submitted.
Jointly with G. Ivanov.
Best and random approximation of a convex body by a polytope
arXiv: 2111.07306, to appear in J. of Complexity.
Jointly with J. Prochno and C. Schuett.
$L_p$-Steiner quermassintegrals
arXiv: 2110.08659, submitted.
Jointly with Kateryna Tatarko.
Floating bodies and approximation of convex bodies by polytopes,
Probability Surveys 19 (2022).
Affine invariant maps for log-concave functions
arXiv: 2010.10889, J. Geometric Analysis 32 (2022).
Jointly with Ben Li and C. Schuett.
Blaschke-Santalo inequality for many functions and geodesic barycenters of measures
arXiv:2010.00135, Advances in Math. 396 (2022).
Jointly with A. V. Kolesnikov.
Convex Floating Bodies of Equilibrium
arXiv:2010.09006, Proceedings of the AMS 150 (2022).
Jointly with D.I. Florentin, C. Schuett, N. Zhang.
Surface area deviation between smooth convex bodies and polytopes
arXiv:1811.04656, Advances in Applied Mathematics 129 (2021).
Jointly with J. Grote and Ch. Thaele.
The convex hull of random points on the boundary of a simple polytope
arXiv: 1911.05917, to appear in J. Discrete & Computational Geometry
Jointly with M. Reitzner and C. Schuett.
Pinsker inequalities and related Monge-Ampere
equations for log concave functions
arXiv:2005.07055, to appear in Indiana Univ.Math.J.
Jointly with U. Caglar, A. V. Kolesnikov.
Asymptotic normality for random simplices and convex bodies in high
dimensions
arXiv: 1906.02471. Proceedings AMS 149 (2021).
Jointly with D.Alonso-Gutierrez, F.Besau, J.Grote, Z.Kabluchko, M.Reitzner, C.Thaele, B.Vritsiou.
The Loewner function of a log-concave function
arXiv: 1904.01211. Journal of Geometric Analysis 31 (2021).
Jointly with Ben Li and Carsten Schuett.
Constrained convex bodies with extremal affine surface areas
arXiv: 1908.07897, Journal of Functional Analysis 279 (2020).
Jointly with O. Giladi, H. Huang, C. Schuett
A Steiner formula in the $L_p$ Brunn Minkowski theory
arXiv:1811.07225, Advances in Math. 355 (2019).
Jointly with Kateryna Tatarko.
Duality of Floating and Illumination Bodies,
arXiv:1709.02424, Indiana Univ. Math. J. 69 (2020).
Jointly with Olaf Mordhorst.
Floating and Illumination Bodies for Polytopes: Duality results,
arXiv:1709.02429, Journal of Discrete Analysis 11, (2019).
Jointly with Olaf Mordhorst.
Data depth and floating body,
arXiv:1809.10925, Statistics Surveys 13 (2019), 52-118.
Jointly with Stanislav Nagy and Carsten Schuett.
Ulam Floating bodies,
arXiv:1803.08224, Journal of London Math. Society 100 (2019).
Jointly with Han Huang and Boaz Slomka.
Floating Functions,
arXiv: 1711.11088, Israel J. Math.231 (2019).
Jointly with Ben Li and Carsten Schuett.
Flag Numbers and Floating Bodies,
arXiv:1805.11287, Advances in Math. 338 (2018).
Jointly with Florian Besau and Carsten Schuett.
Approximation of smooth convex bodies by random polytopes,
Electronic Journal of Probability 23, (2018).
Jointly with Julian Grote.
The Floating Body in Real Space Forms,
Journal Diff. Geometry, Vol. 110, No. 2 (2018), 187-220.
Jointly with Florian Besau.
Weighted floating bodies and polytopal approximation,
TAMS 370, (2018), 7129--7148
Jointly with F. Besau and M. Ludwig.
On the geometry of projective tensor products,
Journal of Functional Analysis 273, (2017), 471--495
Jointly with O. Giladi, J. Prochno, C. Schuett and N. Tomczak-Jaegermann.
Stability results for some geometric inequalities and their functional versions,
The IMA Volumes in Mathematics and its Applications 161:
Convexity and Concentration, Springer (2017)
Jointly with Umut Caglar.
The Spherical Convex Floating Body
Advances in Mathematics 301, 867-901 (2016)
Jointly with Florian Besau.
The Surface Area Deviation of the Euclidean Ball and a Polytope
Journal of Theoretical Probability 31, 244--267 (2018)
Jointly with Steven Hoehner and C.Schuett.
Mixed f-divergence for multiple pairs of measures
Canadian Mathematical Bulletin DOI 10.4153/CMB-2016-050-3 (2016)
Jointly with Deping Ye.
Affine invariant points
Israel J. Math. 208, 163-192 (2015)
Jointly with M. Meyer and C. Schuett.
Dual affine invariant points
Indiana J. Math. 64, 735-768 (2015)
Jointly with M. Meyer and C. Schuett.
Functional versions of L_p-affine surface area and entropy inequalities
IMRN, doi:10.1093/imrn/rnv151 (2015)
Jointly with U.Caglar, M.Fradelizi, O.Guedon, J.Lehec and C.Schuett.
Mixed f-divergence and inequalities for log concave functions
Proc. London Math. Soc. doi: 10.1112/plms/pdu055, (2014)
Jointly with Umut Caglar.
Divergence for s-concave and log concave functions
Advances in Mathematics 257, 219-247 (2014)
Jointly with Umut Caglar.
f-Divergence for convex bodies
Proceedings of the "Asymptotic Geometric Analysis" workshop,
Fields Institute, Toronto (2012).
On the approximation of a polytope by its dual Lp-centroid bodies
Indiana Univ. Math. J. 62, 235-247 (2013)
Jointly with G. Paouris.
Relative entropies for convex bodies
TRANSACTIONS OF THE AMS 366, 2889--2906 (2014)
Jointly with J. Jenkinson.
Renyi Divergence and Lp-affine surface area for convex bodies
Advances in Mathematics 230, 1040--1059 (2012).
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.
Relative entropy of cone measures and Lp centroid bodies
PROCEEDINGS LONDON MATH. SOC. 104 (2012), no. 2, 253–286.
Jointly with G. Paouris.
A note on Mahler's conjecture
INTERNATIONAL MATH. RESEARCH NOTICES, (2012), no. 1, 1–16.
Jointly with S. Reisner and C. Schuett.
Uniform estimates for order statistics and Orlicz functions
POSITIVITY, 16 (2012), no. 1, 1–28.
Jointly with Y. Gordon, A. Litvak and C. Schuett.
On the Homothety Conjecture
Indiana Univ. Math. J. 60 No. 1, 1–20, (2011)
Jointly with Deping Ye.
New affine measures of symmetry for convex bodies
ADVANCES IN MATHEMATICS 228, 2920–2942 (2011)
Jointly with M. Meyer and C. Schuett.
How often is a random quantum state k-entangled
J. PHYSICS A: MATH. THEOR. 44 (2011)
Jointly with S. Szarek and K. Zyczkowski.
Hastings' additivity counterexample via Dvoretzky's theorem
COMMUNICATIONS IN MATHEMATICAL PHYSICS 305 (2011)
Jointly with G. Aubrun and S. Szarek.
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.
Inequalities for mixed p-affine surface area
MATHEMATISCHE ANNALEN 347, 703-737 (2010)
Jointly with Deping Ye.
New Higher Order Equiaffine Invariants
ISRAEL JOURNAL OF MATHEMATICS 171, 221-235 (2009)
Jointly with A. Stancu.
Bipartite states of low rank are almost surely entangled
J. PHYS. A: MATH. THEOR. 42 (2009) 095303
Jointly with M. B. Ruskai.
New Lp affine isoperimetric inequalities
ADVANCES IN MATHEMATICS 218, 762-780 (2008)
Jointly with Deping Ye.
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.
On Lp affine surface areas
INDIANA UNIVERSITY MATH. J. 56, 2305-2324 (2007).
On the minimum of several random variables
PROCEEDINGS AMERICAN MATHEMATICAL SOCIETY 134, 3665-3675 (2006)
Jointly with Y. Gordon, A. Litvak and C. Schuett.
Floating bodies and Illumination bodies
PROCEEDINGS OF THE CONFERENCE "INTEGRAL GEOMETRY AND CONVEXITY"
Wuhan 2004, World Scientific, Singapore (2006).
Approximation of the Euclidean ball by a polytope
STUDIA MATHEMATICA 173, 1-18 (2006)
Jointly with M. Ludwig and C. Schuett.
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.
Surface bodies and p - affine surface area
ADVANCES IN MATHEMATICS 187, 98-145 (2004)
Jointly with C. Schuett.
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.
Orlicz norms of sums of random variables
ANNALS OF PROBABILITY 30 , 1833-1853 (2002)
Jointly with Y. Gordon, A. Litvak and C. Schuett.
The p - affine surface area and geometric interpretations
RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO (2) Suppl. 70, part II, 367-382 (2002)
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.
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.
Dropping a vertex or a facet from a convex polytope
FORUM MATHEMATICUM 13, 359-378 (2001)
Jointly with S. Reisner and C. Schuett.
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.
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.
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.
On the p-affine surface area
ADVANCES IN MATHEMATICS 152, 288-313 (2000)
Jointly with M. Meyer.
A general geometric construction of affine surface area
STUDIA MATHEMATICA 132, 227-238 (1999).
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.
The Santalo-regions of a convex body
TRANSACTIONS OT THE AMS 350, 4569-4591 (1998)
Jointly with M. Meyer.
The illumination body of almost polygonal bodies
GEOMETRIAE DEDICATA 64, 343-354 (1997).
The convex floating body
MATH. SCAND. 66 275-290 (1990)
Jointly with C. Schuett.
Projects
2000 - 2024 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
current
Stephanie Egler
2018 - 2020
Victor Glasgo
Thesis: "Some structural results for convex bodies:
Gravitational Illumination bodies and stability of floating bodies"
2014 - 2018
Ben Li
Thesis: "Convex analysis and its application to quantum information theory"
2014 - 2016 Steven Hoehner ,
Thesis: "The surface area deviation of the Euclidean ball and a polytope"
2011 - 2014 Umut Caglar
Thesis: "Divergence and Entropy inequalities for log concave functions"
2009 - 2013 Justin Jenkinson
Thesis: "Convex Geometric Connections to Information Theory"
2006 - 2009 Deping Ye
Thesis: "Topics in Convex Geometry and Phenomena in High Dimension"
2001 - 2003 Mark Meckes
Thesis: "Random Phenomena in Finite Dimensional Normed Spaces"