Given a convex function $f : \mathbb{R}^n \to [0,\infty)$, let $C$ be a sublevel set of $f$, e.g. $C = \left\lbrace x \in \mathbb{R}^n \mid f(x) \leq 1 \right\rbrace$. Its easy to prove that $C$ is a convex set.
Given an ellipsoid $E = (Q,c)$ such that for $a > 0$ $$ C \subseteq a \cdot (E - c) + c$$ i.e. enlarging $E$ by $a$, will yield an ellipsoid which contains $C$.
The problem is that we want to find a point $x \in C$ such that it maximizes the following term: $$ \left\| L(x-c)\right\|_2$$ where $L$ is the Cholesky decomposition of $Q$ where $Q$ is the matrix which represents the ellipsoid.
This function is NP-Hard since this problem is non-convex optimization problem, also there might be more than one local maximum rather than exactly one as in convex optimization problems.
Lately I have stumbled upon a presentation where they tried to maximize a convex function over a convex set (specifically polytope), using the maximum inscribed ellipsoid: Maximizing Norm
I wonder whether is it possible to apply similar methods or is there a polynomial algorithm for solving the problem and guaranteeing a global maximum?