By how much does the volume of N-dimentional spehere of diameter 1 change if N is increased by 1?

Answer 1

It depends on what value of N you are starting with. The volume of a hypersphere in N dimensions with radius r is given by:

V(r, N) = [pi^(N/2) * r^N]/Gamma(1 + n/2)

where

Gamma(z) is the Euler gamma function, which is a generalization of the factorial function to imaginary and continuous-valued arguments. In this case, if N is even, then Gamma(1 + N/2) = (N/2)!, while if N is odd, Gamma(1 + N/2) = sqrt(pi)*(N!!)/2^((N+1)/2). N!! is the double factorial, and for odd N, N!! =- 1*3*5*7*9..*N

The volumes of hyperspheres with "radius" 1/2, for various values of N up to 10 are:

N=1 V = 1 (i.e., a line segment from -0.5 to +0.5)
N=2, V=0.7854 (i.e., area of a circle with r=0.5)
N=3, V = 0.5236 (i.e., volume of a normal sphere with r=0.5)
N=4, V=0.3084
N=5, V=0.1645
N=6, V=0.0807
N=7, V=0.0369
N=8, V= 0.0159
N=9, V= 6.44*10^-3
N=10, V=2.49*10^-3