Clearly we need to define a probability distribution of points to answer the question.
Perhaps the most interesting describution for which the answer is known is the uniform distribution on the unit sphere . The results of Buchta et al [BMT85] show that the expected number of facets is assymtotically with . The important fact is that it depends linearly on essentially. Here the function is defined recursively by
2 | 1 |
3 | 2 |
4 | 6.76773 |
5 | 31.7778 |
6 | 186.738 |
7 | 1296.45 |
8 | 10261.8 |
9 | 90424.6 |
10 | 872190. |
11 | 9.09402E+06 |
12 | 1.01518E+08 |
13 | 1.20414E+09 |
14 | 1.50832E+10 |
15 | 1.98520E+11 |