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

beatricalwu 2022-07-17 Answered
What is the expected volume of the simplex formed by n + 1 points independently uniformly distributed on S n 1 ?
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Answers (1)

uavklarajo
Answered 2022-07-18 Author has 17 answers
Step 1
More 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 2
In 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 .
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