Wendel's theorem

In geometric probability theory, Wendel's theorem, named after James G. Wendel, gives the probability that N points distributed uniformly at random on an $(n-1)$-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

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

The statement is equivalent to $p_{n,N}$ 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.

This is essentially a probabilistic restatement of Schläfli's theorem that $N$ hyperplanes in general position in $\mathbb {R} ^{n}$ divides it into $2\sum _{k=0}^{n-1}{\binom {N-1}{k}}$ regions.