Some geometric applications of the beta distribution

by: Peter Frankl, Hiroshi Maehara

Annals of the Institute of Statistical Mathematics, Vol. 42, No. 3. (1990), pp. 463-474.

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Abstract

Let ? be the angle between a line and a “random” k-space in Euclidean n-space Rn. Then the random variable cos2 ? 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.

@article{citeulike:1681228, abstract = {Let ? be the angle between a line and a “random” k-space in Euclidean n-space Rn. Then the random variable cos2 ? 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.}, author = {Frankl, Peter and Maehara, Hiroshi }, citeulike-article-id = {1681228}, doi = {10.1007/BF00049302}, journal = {Annals of the Institute of Statistical Mathematics}, keywords = {diameter}, number = {3}, pages = {463--474}, posted-at = {2007-09-21 02:02:38}, priority = {2}, title = {Some geometric applications of the beta distribution}, url = {http://dx.doi.org/10.1007/BF00049302}, volume = {42}, year = {1990} }

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TY - JOUR ID - citeulike:1681228 L3 - citeulike-article-id:1681228 TI - Some geometric applications of the beta distribution JF - Annals of the Institute of Statistical Mathematics VL - 42 IS - 3 SP - 463 EP - 474 N2 - Let ? be the angle between a line and a “random” k-space in Euclidean n-space Rn. Then the random variable cos2 ? 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. KW - diameter AU - Frankl, Peter AU - Maehara, Hiroshi PY - 1990//01/ UR - http://dx.doi.org/10.1007/BF00049302 ER -

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