Wendel's theorem

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help by introducing citations to additional sources. Find sources:  –  ·  ·  · scholar · JSTOR (June 2010)

In geometric probability theory, Wendel's theorem, named after James G. Wendel, gives the probability that N points distributed uniformly at random on an ${\displaystyle (n-1)}$-dimensional hypersphere all lie on the same "half" of the hypersphere. In other words, one seeks the probability that there is some half-space with the origin on its boundary that contains all N points. Wendel's theorem says that the probability is^[1]

${\displaystyle p_{n,N}=2^{-N+1}\sum _{k=0}^{n-1}{\binom {N-1}{k}}.}$

The statement is equivalent to ${\displaystyle p_{n,N}}$ being the probability that the origin is not contained in the convex hull of the N points and holds for any probability distribution on Rⁿ that is symmetric around the origin. In particular this includes all distribution which are rotationally invariant around the origin.

References[]