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

Author(s): Lesław Skrzypek
Journal: Proc. Amer. Math. Soc. 137 (2009), 2661-2669.
MSC (2000): Primary 41A65, 41A44, 42C10
Posted: 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: 10.1090/S0002-9939-09-09875-X
PII: S 0002-9939(09)09875-X
Keywords: Minimal projections, Rademacher projection
Received by editor(s): October 9, 2008
Posted: February 25, 2009
Communicated by: Nigel J. Kalton
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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