TITLE: Sets of evenly distributed points in the Euclidean plane
QUESTION [14 upvotes]: 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?

REPLY [5 votes]: Your second problem is almost the same as the following old, open problem of Danzer and Rogers:
"What is the area of the largest convex region not containing in its interior any one of $n$ given points in a unit square?" Here the big question is whether the answer is $\Theta(\frac 1n)$ or not.
If the answer to the DR-problem is $\omega(\frac 1n)$, then there can be no bound in your problem for the number of points a unit convex set might contain. To see this, suppose by contradiction that you have a $P$ with some $C_P$ bound. Take a $\sqrt n  \times \sqrt n$ size square, this contains $\Theta(n)$ points of $P$, to simplify calculations I suppose it contains exactly $n$. By scaling, the DR-problem gives us an empty convex set of size $\omega(1)$, which becomes bigger than $1$ if $n$ is big enough.
In the other direction, I am not sure if the implication holds, so I think your question is an excellent research topic. Can we prove that the two problems are equivalent?