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Wendel's theorem

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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 ) {\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

p n , N = 2 − N + 1 ∑ k = 0 n − 1 ( N − 1 k ) . {\displaystyle p_{n,N}=2^{-N+1}\sum _{k=0}^{n-1}{\binom {N-1}{k}}.}

The statement is equivalent to p n , N {\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 R 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 {\displaystyle N} hyperplanes in general position in R n {\displaystyle \mathbb {R} ^{n}} divides it into 2 ∑ k = 0 n − 1 ( N − 1 k ) {\displaystyle 2\sum _{k=0}^{n-1}{\binom {N-1}{k}}} regions.

References

  1. Wendel, James G. (1962), "A Problem in Geometric Probability", Math. Scand., 11: 109–111
  2. Cover, Thomas M.; Efron, Bradley (February 1967). "Geometrical Probability and Random Points on a Hypersphere". The Annals of Mathematical Statistics. 38 (1): 213–220. doi:10.1214/aoms/1177699073. ISSN 0003-4851.
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