Wendel's theorem

{{one source|date=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 $(n-1)$-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{{citation

|last = Wendel|first = James G.|title = A Problem in Geometric Probability|journal = Math. Scand.|volume = 11|year = 1962|pages = 109–111| doi=10.7146/math.scand.a-10655 |url = http://www.mscand.dk/article/view/10655}}

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

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 {{math|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 $\backslash R^n$ divides it into $2\backslash sum\_\{k=0\}^\{n-1\}\backslash binom\{N-1\}\{k\}$ regions.{{Cite journal |last1=Cover |first1=Thomas M. |last2=Efron |first2=Bradley |date=February 1967 |title=Geometrical Probability and Random Points on a Hypersphere |journal=The Annals of Mathematical Statistics |volume=38 |issue=1 |pages=213–220 |doi=10.1214/aoms/1177699073 |issn=0003-4851|doi-access=free }}