1.19 (

) Consider a sphere of radius a in D-dimensions together with the concentric hypercube of side 2a, so that the sphere touches the hypercube at the centres of each of its sides. By using the results of Exercise 1.18, show that the ratio of the volume of the sphere to the volume of the cube is given by volume of sphere volume of cube

= πD/2 D2D−1Γ(D/2) . (1.145)

Now make use of Stirling’s formula in the form

Γ(x + 1)  (2π)1/2e

−xxx+1/2 (1.146)

which is valid for x 1, to show that, as D → ∞, the ratio (1.145) goes to zero.

Show also that the ratio of the distance from the centre of the hypercube to one of the corners, divided by the perpendicular distance to one of the sides, is

√

D, which therefore goes to ∞ as D → ∞. From these results we see that, in a space of high dimensionality, most of the volume of a cube is concentrated in the large number of corners, which themselves become very long ‘spikes’!