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On the $ L_p$ norm of the Rademacher projection and related inequalities


Author: Lesław Skrzypek
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
Published electronically: February 25, 2009
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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

$\displaystyle \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,

$\displaystyle \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

DOI: https://doi.org/10.1090/S0002-9939-09-09875-X
Keywords: Minimal projections, Rademacher projection
Received by editor(s): October 9, 2008
Published electronically: February 25, 2009
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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