Projecting the surface measure of the sphere of lnp

We prove that the total variation distance between the cone measure and surface measure on the sphere of l(p)(n) is bounded by a constant times 1/rootn. This is used to give a new proof of the fact that the coordinates of a random vector on the l(p)(n) sphere are approximately independent with density proportional to exp(-\t\(p)), a unification and generalization of two theorems of Diaconis and Freedman. Finally, we show in contrast that a projection of the surface measure of the l(p)(n) in sphere onto a random k-dimensional subspace is "close" to the k-dimensional Gaussian measure. (C) 2003 Editions scientifiques et medicales Elsevier SAS.