Projecting the surface measure of the sphere of ℓ n p

Assaf Naor, Dan Romik

Research output: Contribution to journalArticlepeer-review


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/ √n. 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 sphere onto a random k-dimensional subspace is "close" to the k-dimensional Gaussian measure.

Original languageEnglish (US)
Pages (from-to)241-261
Number of pages21
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number2
StatePublished - 2003

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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