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?

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 $v\in\mathcal C$

1. a convex relation $f(u,v)\leq0$ and

2. linear inequality condition $B(u,v)^T\leq b$ (note $(u,v)^T$ is transpose of vector of $u$ and $v$ variables)

hold where $f:\mathbb R^{2n}\rightarrow\mathbb R$ is a convex polynomial of degree $2$ and $m$ terms and $B\in\mathbb R^{m'\times2n}$ and $b\in\mathbb R^{m'}$ are known. Is it possibly to do this in $O(poly(ntmm'))$ time?

Is it possible to do this at least in trivial case of $f\equiv$ constant.