Elisabeth Werner

A picture of Professor Werner.

Department of Mathematics

Case Western Reserve University

Cleveland, OH 44106

Telephone: (216) 368-2901

elisabeth.werner@case.edu

 

  • Curriculum Vitae
  • Teaching
  • Research Topics
    • Convex Geometry
    • Analysis
    • Probability
  • Numberphile
  • Invited Plenary Speaker, Biannual meeting of the Austrian Mathematical Society, September 2023, Graz (Austria)
  • Simons Fellowship 2020-2021
  • Trimester Programs, Hausdorff Institut

    • Synergies between modern probability, geometric analysis and stochastic geometry, Bonn, Germany, January 2024-April 2024

    • The Interplay between High dimensional Geometry and Probability, Bonn, Germany, January 2021-April 2021

  • Conference Organisation

    • Workshop "Approximation, Geometry and Probability in High Dimensions", Bedlewo, Poland, September 2025.

    • Workshop "Convex Geometry and its Applications", Oberwolfach, Germany, December 2024.

    • Special Session: Probabilistic and Analytic Aspects in Convexity,
      AMS meeting University at Albany, October 2024

    • Workshop "High dimensional phenomena: geometric and probabilistic aspects", Hausdorff Institute Bonn, Germany March 2024

    • Winter School on "Geometry and Probability", Hausdorff Institute Bonn, Germany, January 2024

    • Mini-Symposium "Probabilistic Methods in Convexity" at the Meeting of the Austrian Mathematical Society, Graz, Austria, September 2023

    • 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

    • Archimedes' Principle of floatation and floating bodies: Construction, Extensions and Applications.
      submitted.
      Jointly with Chunyan Liu, Deping Ye and Ning Zhang.

    • Random approximation of convex bodies in Hausdorff metric,
      submitted.
      Jointly with J. Prochno, C. Schuett and M. Sonnleitner.

    • Weighted floating functions and weighted functional affine surface areas,
      submitted.
      Jointly with C. Schuett, Ch. Thaele and N. Turchi.

    • Extremal affine surface areas in a functional setting,
      submitted.
      Jointly with Stephanie Egler.

    • Floating Bodies and Duality in Spaces of Constant Curvature,
      submitted.
      Jointly with F. Besau.

    • Geometric representation of classes of concave functions and duality,
      arXiv:2112.13881, Journal of Geometric Analysis 10.1007/s12220-024-01703-9 (2024)
      Jointly with G. Ivanov.

    • Spherical convex hull of random points on a wedge,
      Mathematische Annalen DOI: 10.1007/s00208-023-02704-9 (2023)
      Jointly with Florian Besau, Anna Gusakova, Matthias Reitzner, Carsten Schuett, Christoph Thaele.

    • Ulam floating functions,
      Geometric Analysis 33 (2023)
      Jointly with Chunyan Liu, Deping Ye, and Ning Zhang.

    • Affine Surface Area,
      Harmonic Analysis and Convexity (De Gruyter) 427-444 (2023)
      Jointly with Carsten Schuett.

    • $L_p$-Steiner quermassintegrals
      arXiv: 2110.08659, Advances in Math. 430 (2023)
      Jointly with Kateryna Tatarko.

    • Curvature functionals on convex bodies,
      Canadian Mathematical Bulletin 66 (2023)
      Jointly with Kateryna Tatarko.

    • The convex hull of random points on the boundary of a simple polytope
      arXiv: 1911.05917, J. Discrete & Computational Geometry 69 (2023)
      Jointly with M. Reitzner and C. Schuett.

    • Best and random approximation of a convex body by a polytope
      arXiv: 2111.07306, J. of Complexity 71 (2022)
      Jointly with J. Prochno and C. Schuett.

    • Pinsker inequalities and related Monge-Ampere equations for log concave functions
      arXiv:2005.07055, Indiana Univ.Math.J. Vol. 71, No. 6 (2022)
      Jointly with U. Caglar, A. V. Kolesnikov.

    • 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.

    • 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 - 2025 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 Diliya Yalikun

    • 2020 - 2023 Stephanie Egler
      Thesis: "Maximal affine surface areas in a functional setting"

    • 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"