Given a compact bounded convex set $\mathcal C\subseteq\mathbb R^n$ given by $t$ hyperplane inequalities I want to find a point $u\in\mathcal C$ such that for all $v\in\mathcal C$ a convex relation $f(u,v)\leq0$ holds where $f:\mathbb R^{2n}\rightarrow\mathbb R$ is a convex polynomial of degree $2$ and $m$ terms. Is it possibly to do this in $O(poly(ntm))$ time?