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Annals of the Institute of Statistical Mathematics

, Volume 42, Issue 3, pp 463–474 | Cite as

Some geometric applications of the beta distribution

  • Peter Frankl
  • Hiroshi Maehara
Geometric Application

Abstract

Let θ be the angle between a line and a “random” k-space in Euclidean n-space Rn. Then the random variable cos2 θ has the beta distribution. This result is applied to show (1) in Rnthere are exponentially many (in n) lines going through the origin so that any two of them are “nearly” perpendicular, (2) any N-point set of diameter d in Rnlies between two parallel hyperplanes distance 2d{(log N)/(n-1)}1/2 apart and (3) an improved version of a lemma of Johnson and Lindenstrauss (1984, Contemp. Math., 26, 189–206). A simple estimate of the area of a spherical cap, and an area-formula for a neighborhood of a great circle on a sphere are also given.

Key words and phrases

Beta distribution Spherical cap Johnson-Lindenstrauss Lemma 

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Copyright information

© The Institute of Statistical Mathematics 1990

Authors and Affiliations

  • Peter Frankl
    • 1
  • Hiroshi Maehara
    • 2
  1. 1.CNRSParisFrance
  2. 2.College of EducationRyukyu UniversityNishihara, OkinawaJapan

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