Minimum volume ellipsoid in $\mathbb{R}^3$ containing all points $(x_1,y_1,z_1),...,(x_n,y_n,z_n)$ formulated as optimization problem

Question

Let $(x_1,y_1,z_1),...,(x_n,y_n,z_n)\in\mathbb{R}^3$. An ellipsoid in $\mathbb{R}^3$ with minimum volume, containing all these points is to be determined.

I am to formulate this as an optimization problem, so I would appreciate it, if someone would look over my attempt and point out any errors:

Let $(x_0,y_0,z_0)$ be the centre of the ellipsoid.
$\text{find}\\ \min V s.t. \frac{4}{3}(x_0+x_i)(y_0+y_i)(z_0+z_i)\leq V \text{ with } i\in\{1,...,n\}$
To my understanding, the LHS gives us an ellipsoid for any of the above mentioned points and our V is the upper bound for these values, therefore the minimum volume.

Answer 1

Hint: Consider the simpler question of minimum bounding sphere: minimize $r$ subject to $$(x_i-x_0)^2+(y_i-y_0)^2+(z_i-z_0)^2 \le r^2.$$ A correct formulation for your problem should simplify to this when the ellipsoid is required to be a sphere.