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Given a convex function $f : \mathbb{R}^n \to [0,\infty)$. Let $L = \left\lbrace x \in \mathbb{R}^n \mid f(x) \leq 1 \right\rbrace$ be it's sub-level set and suppose that $L$ is not empty as well as having a volume of at least $\epsilon > 0$.

Let $E$ be an ellipsoid, such that it's partially contained in $L$, and is represented by a positive semidefinite matrix $G \in \mathbb{R}^{n \times n}$.

We aim in approximating the following:

$$ \max_{x \in L} x^T G x$$

Is there any paper which addresses such problem or have given an approximation towards solving the problem?

Please advise and thanks in advance.

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