If given a convex, compact set $A \subset \mathbb{R}^n$ with finite volume having a minimum cardinality $\epsilon$-net denoted by $\mathcal{N}_\epsilon$, is it possible to find a continuous probability distribution over $A$ such that sampling $O(|\mathcal{N}_\epsilon|)$ points from the distribution would give an $\epsilon$-net or an "almost" $\epsilon$-net (in some sense) with high probability?
Any work pointing in a similar direction would be helpful.