Task: select $n$ random points in a disc. The only requirement is that all points have an equal probability of being chosen.
What I have done: 1) selecting a random real in $[0,2\pi)$ for the angle, and 2) selecting a random radius in $[0,1]$ proportional to the square root to ensure the points don't cluster incorrectly. Such a distribution will give an equal chance for every point to be chosen. I'm satisfied.
Now suppose we wish to plot these points for someone to use. The problem is if I pick, say, 20 points, there is a chance that I will select a random point in the disc that is "too close" to a previously chosen point and the little dots on the plot will overlap. So I keep track of points chosen so far and place additional constraints that new points must be a certain distance from an already chosen point.
Clearly if the distance requirement between points is too large than after the first point, no more points can be chosen. But in practice the gap is not very large relative to the number of points required and there's no worry of having some kind of pathological configuration of points with the values used in practice. Consider, for an idea of how much slack we have, that a thousand points would fit on the diameter line and we only need 25 random points total over the whole disc.
My question is:
Can we say, prior to any points being chosen, that all points have an equal chance of being selected? Does this gap constraint ruin the prior probabilities?
Intuitively I'd say no, the prior probabilities are not ruined, because all points have an equal chance of being the first point, and what happens after that is irrelevant, but I have no background in probability and statistics to rely on to prove this. Against my intuition is the idea that points on the edge become less likely to be chosen due to the gap constraint and that means that points no longer have a uniform chance of being chosen, which was the original requirement.
I know the requirements here are kind of vague but I'm still hoping there's an answer.
Here's a visual suggestion that the answer is "no", the distribution does not remain uniform.
This is an image of many sets (an overlay, if you will) of 15 points in a disc of radius 150, uniformly, with the condition that no two points in a set are closer than 60. You can see the concentration toward the edge, suggesting non-uniformity.
This supports the idea that, since points near the edge eliminate less space for future points, there should be higher concentration of points near the edge.
Not that the picture proves anything, of course.