Conic sections in high dimensions

Question

Can every $n$-dimensional ellipsoid be obtained as a (spherical) conic section?

This is false for generic quadrics but seems true for ellipsoid.

Does anybody have any references?

Answer 1

The answer is no. A spherical cone is $$x_0^2=x_1^2+\ldots+x_n^2.$$ Intersecting it with a hyperplane $x_0=c^Tx$, where $c$ is a (column) vector we obtain a quadratic form with martix $$I-cc^T,$$ that is a rank $1$ perturbation of the unit matrix. The eigenvalues are found from $$0=\det(\lambda I-I+cc^t)=\det((\lambda-1)I+cc^T),$$ so $1-\lambda$ are eigenvalues of $cc^T$. But rank of $cc^T$ is $1$, so $cc^T$ can have at most one non-zero eigenvalue.