Assume you are given the following ellipsoid in $R^n$:
$E: (c+\sum_{i=1}^n \alpha_ix_i)^2$,
where $x_i$ 's are the coordinate variables. c and $\alpha_i$'s are constant.
now the question is how to find a tetrahedron in this space to circumscribe(bound) this ellipsoid E. The tetrahedron has n+1 vertices. so we are looking for n+1 points to surround the E. of course this may have many solutions but one is enough.