Maximum Volume of Unit Radius Sphere in Fractional Dimensions

Theorem
The volume of a unit sphere in $x$-dimensional Euclidean space for real $x$ occurs when $x$ is given as:
 * $x = 5 \cdotp 25694 \, 64048 \, 60 \ldots$

The corresponding volume at that dimension is given by:
 * $V = 5 \cdotp 27776 \, 80211 \, 13400 \, 997 \ldots$