In general in arbitrary dimension, what's the shortest distance from a point inside of an ellipsoid to its surface?
Any good resources on this topic would help greatly as well.
Edit: I know there are ways to do this with constrained optimization, but what I'm wondering is if there's a way to do this analytically.
If the point is $p$ and the ellipsoid is $x^T Q x = 1$, you want to minimize $(x-p)^T(x-p)$ subject to $x^T Q x = 1$, where $Q$ is a (symmetric) positive definite matrix. Using a Lagrange multiplier, we take $$ F = (p-x)^T(p-x) + \lambda (x^T Q x -1)$$ Then we want $$\nabla F = 2 (x-p) + 2 \lambda Q x = 0 $$ i.e. $x = (I+\lambda Q)^{-1} p$ where $\lambda$ is chosen so $x^T Q x = p^T (I+\lambda Q)^{-2} Q p = 1$.
EDIT: There should be at least two real solutions of this equation, of which some will make $x = (I+\lambda Q)^{-1} p$ a local minimum of the distance and others make it a local maximum.