I am able to give a proof to the following inequality for convex functions. Most likely this is well known, but I am unable to find a reference. I would appreciate if someone more knowledgeable in the literature of convex analysis could help.
Suppose $P$ is an open bounded convex subset of $\Bbb R^n$ and $f: P \to \Bbb R$ is a convex function such that $\int_P f =0$. Then there exists positive constants $\alpha,\beta>0$ (not dependent on $f$) such that $$-\alpha\inf_P f \leq \int_P |f(x)|dx^n \leq -\beta \inf_P f .$$
EDIT: per Fedja's observation, I removed the unnecessary constants from the ineqilities. Also, it is specified that $P$ has to be open and bounded. Also, in lack of a good reference, if someone can furnish a short proof of the first estimate, that would work for me as well.