We know a sphere, in $\mathbb{R}^3$-space, is uniquely determined by four points that are not coplanar. Similarly, a hypersphere, in $\mathbb{R}^n$-space, is uniquely determined by $n+1$ points that are not cohyperplanar.
I was just wandering if there exists such a result for getting a unique hyper-ellipsoid? Thank you in advance.