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
pn,N=2-N+1
| n-1 | |
| \sum | |
| k=0 |
\binom{N-1}{k}.
The statement is equivalent to
pn,N
This is essentially a probabilistic restatement of Schläfli's theorem that
N
\Rn
| n-1 | |
| 2\sum | |
| k=0 |
\binom{N-1}{k}