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Low dimensional sections versus projections of convex bodies

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Abstract

The structure of low dimensional sections and projections of symmetric convex bodies is studied. For a symmetric convex bodyB ⊂ ℝn, inequalities between the smallest diameter of rank ℓ projections ofB and the largest in-radius ofm-dimensional sections ofB are established, for a wide range of sub-proportional dimensions. As an application it is shown that every bodyB in (isomorphic) ℓ-position admits a well-bounded (√n, 1)-mixing operator.

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References

  1. J. Bourgain and S. J. Szarek,The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization, Israel Journal of Mathematics62 (1988), 169–180.

    MATH  MathSciNet  Google Scholar 

  2. J. Bourgain and L. Tzafriri,Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel Journal of Mathematics57 (1987), 137–224.

    MATH  MathSciNet  Google Scholar 

  3. E. D. Gluskin,Finite-dimensional analogues of spaces without basis, Doklady Akademii Nauk SSSR216 (1981), 1046–1050.

    MathSciNet  Google Scholar 

  4. A. Giannopoulos, V. D. Milman and A. Tsolomitis,Asymptotic formulas for the diameter of sections of symmetric convex bodies, Journal of Functional Analysis223 (2005), 86–108.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Mankiewicz,Finite-dimensional Banach spaces with symmetry constant of order √n, Studia Mathematica79 (1984), 193–200.

    MATH  MathSciNet  Google Scholar 

  6. P. Mankiewicz,Subspace mixing properties of operators in ℝn,with applications to Gluskin spaces, Studia Mathematica88 (1988), 51–67.

    MATH  MathSciNet  Google Scholar 

  7. P. Mankiewicz and N. Tomczak-Jaegermann,Quotients of finite-dimensional Banach spaces; random phenomena, inHandbook of the Geometry of Banach Spaces (W. B. Johnson and J. Lindenstrauss, eds.), Vol. II, Elsevier, Amsterdam, 2003, pp. 1201–1246.

    Chapter  Google Scholar 

  8. P. Mankiewicz and N. Tomczak-Jaegermann,Geometry of families of random projections of symmetric convex bodies, Geometric and Functional Analysis11 (2001), 1282–1326.

    Article  MATH  MathSciNet  Google Scholar 

  9. V. D. Milman,Randomness and Pattern in Convex Geometric Analysis, Proceedings of ICM Berlin, Vol. 2, Documenta Mathematica, 1998, pp. 665–677.

    MathSciNet  Google Scholar 

  10. V. D. Milman,Some applications of duality relations, inGeometric Aspects of Functional Analysis, (1989–90), Lecture Notes in Mathematics1469, Springer, Berlin, 1991, pp. 13–40.

    Google Scholar 

  11. V. D. Milman,A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Functional Analysis and its Applications5 (1971), 28–37 (English translation).

    MathSciNet  Google Scholar 

  12. V. D. Milman and G. Schechtmann,Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics1200, Springer-Verlag, Berlin, 1986.

    MATH  Google Scholar 

  13. V. D. Milman and G. Schechtman,Global versus local asymptotic theories of finite-dimensional normed spaces, Duke Mathematical Journal90 (1997), 73–93.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Pisier,Volumes of Convex Bodies and Banach Space Geometry, Cambridge University Press, 1989.

  15. S. J. Szarek,The finite-dimensional basis problem with an appendix on nets of Grassman manifolds, Acta Mathematica151 (1983), 153–179.

    Article  MATH  MathSciNet  Google Scholar 

  16. S. J. Szarek,On the existence and uniqueness of complex structure and spaces with “few” operators, Transactions of the American Mathematical Society293 (1986), 339–353.

    Article  MATH  MathSciNet  Google Scholar 

  17. S. J. Szarek and N. Tomczak-Jaegermann,Saturating constructions for normed spaces, Geometric and Functional Analysis, to appear.

  18. N. Tomczak-Jaegermann,Banach-Mazur Distances and Finite Dimensional Operator Ideals, Pitman Monographs, Longman, Harlow, 1989.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, PAN, Śniadeckich 8, P.O. Box 21, 00-956, Warsaw 10, Poland

    Piotr Mankiewicz

  2. Department of Mathematical and Statistical Sciences, University of Alberta, T6G 2G1, Edmonton, Alberta, Canada

    Nicole Tomczak-Jaegermann

Authors
  1. Piotr Mankiewicz
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  2. Nicole Tomczak-Jaegermann
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Corresponding author

Correspondence to Piotr Mankiewicz.

Additional information

Research of this author was partially supported by KBN Grant no. 1 P03A 015 27.

This author holds the Canada Research Chair in Geometric Analysis.

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Mankiewicz, P., Tomczak-Jaegermann, N. Low dimensional sections versus projections of convex bodies. Isr. J. Math. 153, 45–60 (2006). https://doi.org/10.1007/BF02771778

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  • DOI: https://doi.org/10.1007/BF02771778

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