On the 𝐿_{𝑝} norm of the Rademacher projection and related inequalities

Skrzypek, Lesław

doi:10.1090/S0002-9939-09-09875-X

On the 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

. 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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