reference request – Approximating a probability density with a point set Answer

plane geometry - Does the intersection of two moving curves sweep out a continuous line?

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reference request – Approximating a probability density with a point set

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.

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