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

Hello dear visitor to our network We will offer you a solution to this question reference request – Approximating a probability density with a point set ,and the answer will be typical through documented information sources, We welcome you and offer you new questions and answers, Many visitor are wondering about the answer to this question.

## 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.

we will offer you the solution to reference request – Approximating a probability density with a point set question via our network which brings all the answers from multiple reliable sources.