Research of Carsten Schütt

Research-Forschung


Meine Forschungsgebiete sind Funktionalanalysis, Approximationstheorie und Konvexgeometrie. In der Funktionalanalysis liegt der Schwerpunkt meiner Forschung in der Geometrie der Banachräume. Hauptsächlich habe ich die Geometrie endlichdimensionaler, normierter Räume untersucht. Häufig betrachtet man hier das Verhalten einer geometrischen Invarianten auf einer bestimmten Klasse von normierten Räumen. Man versucht, das asymptotische Verhalten der geometrischen Invarianten für grosse Dimension der Räume zu bestimmen. Dies führt meistens auf geometrische Ungleichungen in dem Rn. Die Ungleichungen stellen Abschätzungen zwischen verschiedenen Normen dar, oder auch Abschätzungen geometrischer Invarianten von Operatoren, wie z.B. Approximationszahlen und Entropiezahlen. Die Methoden sind geometrischer, kombinatorischer und wahrscheinlichkeitstheoretischer Natur. In dem Gebiet der Konvexgeometrie habe ich die Eigenschaften konvexer Körper im Rn untersucht. So habe ich mich Projektionenkörpern, Affinoberflächen, Schwimmkörpern von konvexen Körpern und Zufallspolytopen gewidmet. Dabei bin ich auch mit Fragestellungen der affinen Differentialgeometrie und der Approximationstheorie in Berührung gekommen. Einen grösseren Raum in meiner Forschung nahm die Approximation konvexer Körper durch Polytope mit einer festen Eckenzahl ein. Ich habe u.a. die beste Approximation mit zufälliger Approximation verglichen.



List of Publications

  1. The projection constants of finite dimensional spaces whose unconditional basis constant is 1, Israel Journal of Mathematics 30 (1978), 207--212.
  2. Unconditionality in tensor product, Israel Journal of Mathematics 31 (1978), 209--216.
  3. Some geometric properties of finite dimensional, symmetric Banach spaces, Seminaire d'Analyse Fonctionelle 1980--81, Ecole Polytechnique.
  4. A remark on the unconditional structure of L(E, F), Israel Journal of Mathematics 39 (1981), 16-24.
  5. (with A. Pelczynski), Factoring the natural injection $i(n):L^{\infty}n \rightarrow L1n $ through finite dimensional Banach spaces and geometry of finite dimensional unitary ideals, Advances in Mathematics Supplementary Studies, vol. 7B, Academic Press, 1981, 653--683.
  6. On the uniqueness of symmetric bases in finite dimensional Banach spaces, Israel Journal of Mathematics 40 (1981), 97--117.
  7. On the Banach-Mazur distance of finite dimensional Banach Spaces and the hypergeometric distribution, Studia Mathematica 72 (1982), 109--129.
  8. On the volume of unit balls in Banach spaces, Compositio Math. 47 (1982), 393--407.
  9. Entropy numbers of diagonal operators between symmetric Banach spaces, Journal of Approximation Theory 40 (1984), 121--128.
  10. Banach spaces of operators and local unconditional structure, Proc. 3rd Paderborn Conf. Funct. Anal., May 24--29, 1983, North-Holland, 47--58.
  11. On the positive projection constant, Studia Mathematica 78 (1984), 185--198.
  12. (with S. Kwapien ), Some combinatorial and probabilistic inequalities and their application to Banach space theory, Studia Mathematica 82 (1985), 91--106.
  13. (with Y. Gordon and H. König ), Geometric and probabilisitic estimates for entropy and approximation number of operators, Journal of Approximation Theory 49 (1987), 219--239.
  14. (with Y. Raynaud), Some results on symmetric subspaces of L1, Studia Mathematica 89 (1988), 27--35.
  15. Lorentz spaces that are isomorphic to subspaces of L1, Transactions of the American Mathematical Society 314 (1989), 583--595.
  16. The isoperimetric quotient and some classical Banach spaces, Israel Journal of Mathematics 67 (1989), 43--61.
  17. (with S. Kwapien ) Some combinatoral and probabilistic inequalities and their application to Banach space theory II, Studia Mathematica 95 (1989), 141--154.
  18. (with E. Werner ) The convex floating body, Mathematica Scandinavica 66 (1990), 275--290.
  19. The convex floating body and polyhedral approximation, Israel Journal of Mathematics 73 (1991), 65--77.
  20. (with E. Werner ) The convex floating body of almost polytopal bodies, Geometriae Dedicata 44 (1992), 169--188.
  21. On the affine surface area, Proceedings of the American Mathematical Society 118(1993), 1213--1218.
  22. (with E. Werner ) Homothetic floating bodies, Geometriae Dedicata 49(1994), 335--348
  23. Random polytopes and affine surface area, Mathematische Nachrichten 170(1994), 227--249
  24. On the embedding of 2-concave Orlicz spaces into L1, Studia Mathematica 113(1995), 73--80
  25. (with Y. Gordon und S. Reisner ) Umbrellas and polytopal approximation of the Euclidean ball, Journal of Approximation Theory 90(1997), 9--22
  26. Floating body, illumination body, and polytopal approximation, Comptes Rendus de l'Academie des Sciences Paris 324(1997), 201--203
  27. Floating body, illumination body, and polytopal approximation, in: K.M. Ball und V. Milman: Convex Geometric Analysis , Mathematical Sciences Research Institute Publications 34, Cambridge University Press, 1999, 203--230
  28. (with H. König and N. Tomczak-Jaegermann ) Projection constants of symmetric spaces and variants of Khintchine's inequality, Journal für die reine und angewandte Mathematik 511(1999), 1-42
  29. (with E. Werner ) Random polytopes with vertices on the boundary of a convex body, Comptes Rendus de l'Academie des Sciences Paris 331(2000), 697-201
  30. (with P. Mankiewicz) A simple proof of an estimate for the approximation of the Euclidean ball and the Delone triangulation numbers, Journal of Approximation Theory 107(2000), 268-280
  31. (with S. Reisner and E. Werner ) Dropping a vertex or a facet from a convex polytope, Forum Mathematicum 13(2001), 359-378
  32. (with P. Mankiewicz) On the Delone triangulation numbers, Journal of Approximation Theory 111(2001), 139-142
  33. (with Y. Gordon , A. Litvak und E. Werner ) Orlicz norms of sequences of random variables, Annals of Probability 30(2002), 1833--1853
  34. (with Y. Gordon , A. Litvak und E. Werner ) Geometry of spaces constructed between polytopes and related zonotopes, Bulletin des sciences math\'ematiques 126(2002), 733-762
  35. Best and random approximation of convex bodies by polytopes, Rendiconti del Circolo Matematico di Palermo 70 (2002), 315--334
  36. (with E. Werner) Polytopes with vertices chosen randomly from the boundary of a convex body Israel Seminar 2001-2002, Lecture Notes in Mathematics 1807 (V.D. Milman and G. Schechtman, eds.), Springer-Verlag, 2003, 241--422
  37. (with E. Werner) Surface bodies and p-affine surface area, Advances in Mathematics, 187 (2004), 98-145
  38. (with Y. Gordon, A. Litvak and E. Werner) Minima of sequences of Gaussian random variables, C.R. Acad. Sci. Paris, Ser. I, 340 (2005), 445-448
  39. (with Y. Gordon, A. Litvak and E. Werner) On the minimum of several random variables, Proceedings of the AMS, 134(2006), 3665-3675
  40. (with M. Ludwig and E. Werner) Approximation of the Euclidean ball by a polytope, Studia Mathematica,173 (2006), 1--18
  41. Random methods in approximation of convex bodies, Integral Geometry and Convexity, eds. E. Grinberg, S. Li, G. Zhang, J. Zhou, Proceedings of the International Conference, Wuhan, China 18-23 October 2004, World Scientific 2006
  42. (with K. Böröczky, K. Böröczky Jr. and G. Wintsche) Convex bodies of minimal volume, surface area and mean width with respect to thin shells, Canadian Journal of Mathematics, 60 (2008), 3-33.
  43. (with M. Meyer and E. Werner) New affine measures of symmetry for convex bodies, Advances in Mathematics, 228 (2011), 2920-2942.
  44. (with S. Reisner and E. Werner) Mahler's conjecture and curvature, International Mathematics Research Notices, 2012 (2012), 1-16.
  45. (with Y. Gordon, A. Litvak and E. Werner) Uniform estimates for order statistics and Orlicz functions, Positivity, 16 (2012), 1-28.
  46. (with M. Fradelizi and G. Paouris) Simplices in the Euclidean ball, Canadian Mathematical Bulletin, 55 (2012), 498-508.
  47. (with S. Artstein-Avidan, B. Klartag und E. Werner) Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality, Journal of Functional Analysis, 262 (2012), 4181-4204.
  48. (with J. Prochno) Combinatorial inequalities and subspaces of L1, Studia Mathematica, 211 (2012), 21-39.
  49. (with S. Riemer) On the expectation of the norm of random matrices with non-identically distributed entries, Electronic Journal of Probability, 18 (2013), 1-13.
  50. (with M. Meyer and E. Werner) Affine invariant points, Israel Journal of Mathematics, (), .
  51. (with M. Meyer and E. Werner) Dual affine invariant points, Indiana Journal of Mathematics, (), .
  52. (with U. Caglar, M. Fradelizi, O. Guedon, J. Lehec and E. Werner) Functional versions of Lp-affine surface area and entropy inequalities, International Mathematics Research Notices, (), .