Volume of largest ellipsoid fitting into a box and volume of smallest ellipsoid containing that box

Question

We have a box, called B, contianing side lengths $a_1, a_2, a_3, . . . , a_n $. What is the volume of the largest ellipsoid fitting into B, and what's the volume of the smallest ellipsoid containing B?

Answer 1

The volume of the unit $n$-ball, a ball in $n$ dimensions is $\frac {\pi^{\frac n2}}{\Gamma\left(\frac n2 +1\right)}$. This fits nicely inside the side $2$ $n-$hypercube. Now just stretch the axes by $\frac {a_1}2$ for the first axis, $\frac {a_2}2$ for the second and so on. The volume of the $n-$ ellipsoid will be $\frac {\pi^{\frac n2}}{\Gamma\left(\frac n2 +1\right)}\frac 1{2^n}\prod_{i=1}^na_i$

Similarly, the ball that fits around an $n-$hypercube has a diameter of $\sqrt n$ so a volume of $\frac {\pi^{\frac n2}}{\Gamma\left(\frac n2 +1\right)}\frac 1{2^n}n^{\frac n2}$. We can again stretch the axes and get the volume of the circumscribed ellipsoid as $\frac {\pi^{\frac n2}}{\Gamma\left(\frac n2 +1\right)}\frac 1{2^n}n^{\frac n2}\prod_{i=1}^na_i$