Maximum Ball

Geometry Level 2

What is the value of the dimension \(n\) that maximizes the volume of unit \(n\)-ball \[ \left\{ \left( x_1,x_2,\cdots,x_n \right) \in\mathbb{R}^n\, \mid \, x_1^2+x_2^2+\cdots+x_n^2\le 1 \right\}?\]

Hint: The volume of \(n\)-ball of radius \(R\) is \[V_n(R)=\frac{\pi^{\frac n2}}{\Gamma \left(\frac n2+1\right)}R^n,\] where \(\Gamma(\cdot)\) denotes the gamma function.