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

Abstract

We prove a concentration inequality for the ℓ q n \ell _q^n norm on the ℓ p n \ell _p^n sphere for p , q > 0 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 ℓ p n \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 p .