# zbMATH — the first resource for mathematics

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