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

Skrzypek, Lesław

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

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