Using Rademacher permutations to reduce randomness
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- by S. Artstein-Avidan and V. D. Milman
- St. Petersburg Math. J. 19 (2008), 15-31
- DOI: https://doi.org/10.1090/S1061-0022-07-00983-1
- Published electronically: December 12, 2007
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Abstract:
It is shown how a special family of unitary operators, called the Rademacher permutations and related to the Clifford algebra, can be used to reduce the level of randomness in several results in asymptotic geometric analysis.References
- Noga Alon and Joel H. Spencer, The probabilistic method, 2nd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience [John Wiley & Sons], New York, 2000. With an appendix on the life and work of Paul Erdős. MR 1885388, DOI 10.1002/0471722154
- Shiri Artstein-Avidan, A Bernstein-Chernoff deviation inequality, and geometric properties of random families of operators, Israel J. Math. 156 (2006), 187–204. MR 2282375, DOI 10.1007/BF02773831
- S. Artstein-Avidan and V. D. Milman, Logarithmic reduction of the level of randomness in some probabilistic geometric constructions, J. Funct. Anal. 235 (2006), no. 1, 297–329. MR 2216448, DOI 10.1016/j.jfa.2005.11.003
- S. Artstein-Avidan, O. Friedland, and V. Milman, Geometric applications of Chernoff-type estimates and a zigzag approximation for balls, Proc. Amer. Math. Soc. 134 (2006), no. 6, 1735–1742. MR 2204286, DOI 10.1090/S0002-9939-05-08144-X
- —, More geometric applications of Chernoff inequality, Geometric Aspects of Functional Analysis, Springer Lecture Notes in Math. (to appear).
- Gerald H. L. Cheang and Andrew R. Barron, A better approximation for balls, J. Approx. Theory 104 (2000), no. 2, 183–203. MR 1761898, DOI 10.1006/jath.1999.3441
- Aharon Ben-Tal and Arkadi Nemirovski, On polyhedral approximations of the second-order cone, Math. Oper. Res. 26 (2001), no. 2, 193–205. MR 1895823, DOI 10.1287/moor.26.2.193.10561
- J. Bourgain, J. Lindenstrauss, and V. D. Milman, Minkowski sums and symmetrizations, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 44–66. MR 950975, DOI 10.1007/BFb0081735
- Apostolos A. Giannopoulos and Vitali D. Milman, Euclidean structure in finite dimensional normed spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 707–779. MR 1863705, DOI 10.1016/S1874-5849(01)80019-X
- M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), no. 4, 843–854. MR 708367, DOI 10.2307/2374298
- Vitali D. Milman, The concentration phenomenon and linear structure of finite-dimensional normed spaces, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 961–975. MR 934298
- V. D. Milman and A. Pajor, Regularization of star bodies by random hyperplane cut off, Studia Math. 159 (2003), no. 2, 247–261. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish). MR 2052221, DOI 10.4064/sm159-2-6
- Vitali D. Milman and Gideon Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics, vol. 1200, Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. MR 856576
- V. D. Milman and G. Schechtman, Global versus local asymptotic theories of finite-dimensional normed spaces, Duke Math. J. 90 (1997), no. 1, 73–93. MR 1478544, DOI 10.1215/S0012-7094-97-09003-7
- Gilles Pisier, The volume of convex bodies and Banach space geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge, 1989. MR 1036275, DOI 10.1017/CBO9780511662454
Bibliographic Information
- S. Artstein-Avidan
- Affiliation: School of Mathematical Science, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel
- Email: shiri@post.tau.ac.il
- V. D. Milman
- Affiliation: School of Mathematical Science, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel
- MR Author ID: 125020
- ORCID: 0000-0003-4632-5487
- Email: milman@post.tau.ac.il
- Received by editor(s): August 1, 2006
- Published electronically: December 12, 2007
- © Copyright 2007 American Mathematical Society
- Journal: St. Petersburg Math. J. 19 (2008), 15-31
- MSC (2000): Primary 52A21
- DOI: https://doi.org/10.1090/S1061-0022-07-00983-1
- MathSciNet review: 2319508
Dedicated: Dedicated to Professor V. A. Zalgaller on the occasion of his 85th birthday