Projecting the surface measure of the sphere of \({\ell}_p^n\). (English) Zbl 1012.60025

Naor, Assaf; Romik, Dan

Ann. Inst. Henri Poincaré, Probab. Stat. 39, No. 2, 241-261 (2003).

Summary: We prove that the total variation distance between the cone measure and surface measure on the sphere of \({\ell}_p^n\) is bounded by a constant times \(1/\sqrt n\). This is used to give a new proof of the fact that the coordinates of a random vector on the \(\ell_p^n\) sphere are approximately independent with density proportional to exp\((-|t|^p)\), a unification and generalization of two theorems of P. Diaconis and D. Freedman [ibid. 23, Suppl., 397-423 (1987; Zbl 0619.60039)]. Finally, we show in contrast that a projection of the surface measure of the \({\ell}_p^n\) sphere onto a random \(k\)-dimensional subspace is “close” to the \(k\)-dimensional Gaussian measure.

MSC:

60F05	Central limit and other weak theorems
52A20	Convex sets in \(n\) dimensions (including convex hypersurfaces)

Keywords:

cone measure; surface measure; projection