True or false: In a 2D Poisson process, for every point $P$, there exists a convex $1000$-gon with Poisson points as vertices, that contains only $P$.

Question

I made a Desmos graph that generates $30$ uniformly random black points in a disk, with the centre of the disk in red.

I asked myself, "Can I always draw a convex quadrilateral with four of the random points as vertices, such that its interior contains the centre of the disk but does not contain any of the random points?" The answer seems to be yes, based on experimentation.

Then I asked myself the same question, except with "quadrilateral" replaced by pentagon, hexagon, or heptagon, and the answer still seems to be always yes.

So my question is:

True or false: In a 2D Poisson process, for every point $P$, there exists a convex $1000$-gon with Poisson points as vertices, whose interior contains $P$ and none of the other random points.

I suspect the answer is yes. By making the polygon very long and narrow, maybe we can always find enough points to serve as vertices. But I'm not sure if this is true.

Context: Recently I have been investigating inherent properties of the $2D$ Poisson process, for example here, here and here.

Answer 1

(Self-answering. I think I've figured it out.)

The answer is: true.

Given a $2D$ Poisson process and one of the points $P$, consider a square with centre $P$ and horizontal and vertical sides, that encloses exactly $1000$ points in the Poisson process. Call this set of $1000$ points $S_1$.

If the points in $S_1$ are the vertices of a convex $1000$-gon that contains $P$, then we are done. If not, then expand the square until it contains eactly $1000$ points whose $y$-coordinates are all between $-y_1$ and $y_1$, where $y_1$ is absolute value of the $y$-coordinate closest to $0$ among the points in $S_1$. Call this set of $1000$ points $S_2$.

If the points in $S_2$ are the vertices of a convex $1000$-gon that contains $P$, then we are done. If not, then expand the square until it contains eactly $1000$ points whose $y$-coordinates are all between $-y_2$ and $y_2$, where $y_2$ is the absolute value of the $y$-coordinate closest to $0$ among the points in $S_2$. Call this set of $1000$ points $S_3$. And so on.

The probability that $1000$ random points in a rectangle are the vertices of a convex $1000$-gon that contains the rectangle's centre is some positive constant. (The height/width ratio of the rectangle doesn't matter: we can arbitrarily vertically stretch the rectangle without affecting the relative distribution of random points.) So if we repeatedly choose $1000$ random points in a rectangle, with probability $1$ we will eventually get a set of $1000$ points that are the vertices of a convex $1000$-gon that contains the rectangle's centre.

So eventually we will find a convex $1000$-gon that contains $P$ and no other random point.