Let $f$ be a "nice" probability density on $\mathbb{R}^2$, let $p=1/k$ for some fixed positive integer $k$, and let $\epsilon>0$. Are there any known statements of the following form?
"There exists a point set $Q$ of size $n = n(\epsilon)$ such that for every convex subset $C$ that satisfies $|C\cap Q| = pn$, we have $\left| \int_C f(x)~dx - p\right|\leq \epsilon$."
I think it is clear that $n$ would also depend on Lipschitz constants associated with $f$, or its support region, in addition to $\epsilon$, but I cannot find a reference to such statements. I would assume that $n=O(1/\epsilon^2)$ based on basic dimensional intuition, but cannot justify such a claim.
