Some geometric applications of the beta distribution

Let be the angle between a line and a random k-space in Euclidean n-space R ⁿ. Then the random variable cos² has the beta distribution. This result is applied to show (1) in R ⁿthere 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 R ⁿlies 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.