Is there a set $P \subset \mathbb{R}^2$ of points in the Euclidean plane whose intersection with every convex subset of $\mathbb{R}^2$ of area $1$ is nonempty but finite?

If the answer is *yes*, can $P$ be chosen in such way that there is a constant $C_P$ with
the property that for every convex subset $S \subset \mathbb{R}^2$ of area $1$ we have
$1 \leq |S \cap P| \leq C_P$? -- And if yes, which $P$ admit the smallest $C_P$?

*Remarks:*

Lattices are not examples as there is always an $\epsilon > 0$ such that there are $\epsilon \times \frac{1}{\epsilon}$ rectangles which do not contain a lattice point.

The question looks in some sense natural to me, and I wonder whether it has already been considered before. Maybe someone knows a reference?