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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the $L_p$ norm of the Rademacher projection and related inequalities
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by Lesław Skrzypek PDF
Proc. Amer. Math. Soc. 137 (2009), 2661-2669 Request permission

Abstract:

The purpose of this paper is to find the exact norm of the Rademacher projection onto $\{r_1,r_2,r_3\}.$ Namely, we prove \[ \Vert R_3\Vert _p=\frac {(3^{p/q}+1)^{1/p}(3^{q/p}+1)^{1/q}}{4}. \] The same techniques also give the relative projection constant of $\ker \{1,...,1\}$ in $\ell _p^n,$ that is, \[ \lambda (\ker \{1,...,1\},\ell _p^n)=\frac {((n-1)^{p/q}+1)^{1/p}((n-1)^{q/p}+1)^{1/q}}{n}, \] for $n=2,3,4$. We discuss the relation of the above inequalities to the famous Khintchine and Clarkson inequalities. We conclude the paper by stating some conjectures that involve the geometry of the unit ball of $\ell _p^n.$
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Additional Information
  • Lesław Skrzypek
  • Affiliation: Department of Mathematics, University of South Florida, 4202 E. Fowler Avenue, PHY 114, Tampa, Florida 33620-5700
  • Email: skrzypek@math.usf.edu
  • Received by editor(s): October 9, 2008
  • Published electronically: February 25, 2009
  • Communicated by: Nigel J. Kalton
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 2661-2669
  • MSC (2000): Primary 41A65, 41A44, 42C10
  • DOI: https://doi.org/10.1090/S0002-9939-09-09875-X
  • MathSciNet review: 2497479