The surface measure and cone measure on the sphere of ℓ_{𝑝}ⁿ

Naor, Assaf

doi:10.1090/S0002-9947-06-03939-0

The surface measure and cone measure on the sphere of $\ell _p^n$
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Trans. Amer. Math. Soc. 359 (2007), 1045-1079 Request permission

Abstract:

We prove a concentration inequality for the $\ell _q^n$ norm on the $\ell _p^n$ sphere for $p,q>0$. This inequality, which generalizes results of Schechtman and Zinn (2000), is used to study the distance between the cone measure and surface measure on the sphere of $\ell _p^n$. In particular, we obtain a significant strengthening of the inequality derived by Naor and Romik (2003), and calculate the precise dependence of the constants that appeared there on $p$.

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