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:*


1. 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.
2. The question looks in some sense natural to me, and I wonder whether
it has already been considered before. Maybe someone knows a reference?



Sets of evenly distributed points in the Euclidean plane
There is a set $P$. For the construction of this set first take the squares of area $1$ whose edges are integers and numerate them. For each square, say $S\_n$, you can take a square lattice in it such that any convex inside the square that do not intersect the lattice has area less than $a\_n$ for any $a\_n>0$, just take a lattice with a very small distance.


Now choose lattices and the $a\_n$'s so that $\sum\_{n=1}^\infty a\_n<1$.


This works because if you have a convex $C$ in $\mathbb R^2$ then $C\cap S\_n$ is convex, and if it do not intersect the set $P$ then $$m(C)=\sum\_{n=1}^\infty m(C\cap S\_n)\leq\sum\_{n=1}^\infty a\_n<1,$$ where $m(A)$ denotes the area of $A$.


For the second question there is a proof here <http://www.math.tau.ac.il/~barakw/papers/dynamical_gowers.pdf> that there could not be any constant. 


