What is the expected volume of the simplex formed by n + 1 points independently uniformly distributed on S n − 1 ?

Question

What is the expected volume of the simplex formed by   n  +  1 points independently uniformly distributed on             S              n      −      1      ?

uavklarajo · Accepted Answer

Step 1More generally, for i points independently uniformly distributed in the interior of the n-ball and j points independently uniformly distributed on its boundary (the sphere             S              n      −      1      ), with   1  ≤  r  :=  i  +  j  −  1  ≤  n so that the points almost surely form an r-simplex, the moments of the volume   Δ of this simplex are  E      [          Δ      k        ]    =        1          r              !        k                        (              n                  n          +          k                    )        i              Γ              (                  1          2                (        r        +        1        )        (        n        +        k        )        −        j        +        1        )                    Γ              (                  1          2                [        (        r        +        1        )        n        +        r        k        ]        −        j        +        1        )                        (                        Γ                      (                          1              2                        n            )                                    Γ                      (                          1              2                        [            n            +            k            ]            )                              )        r        ∏          l      =      1              r      −      1                  Γ              (                  1          2                [        n        −        r        +        k        +        l        ]        )                    Γ              (                  1          2                [        n        −        r        +        l        ]        )                .Step 2In our case,   i  =  0,   j  =      n    +    1  ,   r  =  n and   k  =  1, so the desired volume is      A    n    =      1          n      !                  Γ              (                  1          2                          n          2                +                  1          2                )                    Γ              (                  1          2                          n          2                )                        (                        Γ                      (                          1              2                        n            )                                    Γ                      (                          1              2                        n            +                          1              2                        )                              )        n        ∏          l      =      1              n      −      1                  Γ              (                  1          2                l        +                  1          2                )                    Γ              (                  1          2                l        )                .With                    Ξ        (        n        )        :=                              Γ                          (              n              +                              1                2                            )                                            Γ            (            n            )                              this becomes      A    n    =      1          n      !        Ξ      (                  n        2            2        )    Ξ            (              n        2            )              −      n            ∏          l      =      1              n      −      1        Ξ      (          l      2        )      .Thus, with                    n                              1          2                            1                              3          2                            2                              9          2                            8                            Ξ        (        n        )                              1                      π                                                            π                    2                                      2                      π                                                            3                          π                                4                                      128                      35                          π                                                                        6435                          π                                4096                    we find      A    2    =      1    2              Ξ      (      2      )      Ξ              (                  1          2                )                    Ξ      (      1      )      Ξ      (      1      )        =      3          2      π      and      A    3    =      1          3      !                  Ξ              (                  9          2                )            Ξ              (                  1          2                )            Ξ      (      1      )              Ξ              (                  3          2                )            Ξ              (                  3          2                )            Ξ              (                  3          2                )              =            4      π        105      ,in agreement with the MathWorld values, and also      A    4    =      1          4      !                  Ξ      (      8      )      Ξ              (                  1          2                )            Ξ      (      1      )      Ξ              (                  3          2                )                    Ξ      (      2      )      Ξ      (      2      )      Ξ      (      2      )      Ξ      (      2      )        =      6435          31104              π        2                .

What is the expected volume of the simplex formed by n+1 points independently uniformly distributed on S^{n-1}?

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