Maximum Ball

Geometry Level 2

What is the value of the dimension nn that maximizes the volume of unit nn-ball {(x1,x2,,xn)Rnx12+x22++xn21}? \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 nn-ball of radius RR is Vn(R)=πn2Γ(n2+1)Rn,V_n(R)=\frac{\pi^{\frac n2}}{\Gamma \left(\frac n2+1\right)}R^n, where Γ()\Gamma(\cdot) denotes the gamma function.

×

Problem Loading...

Note Loading...

Set Loading...