As for the decrease, it is obvious that the minimum distance from the center to points decreases with . Just give it a thought.
I show you how to find the distribution of the minimum distance from the center to points uniformly chosen in a disk.
First thing is to consider only one point uniformly distributed in a disk. I will consider a disk with center and radius . Let (i.e. is the distance from to ).
For , we have of course , and if then (because means that is inside the circle with radius , and the probability to be in a subset of the disk is proportional to its area). Using this computation, you can get the density of the distribution of (by differentiating, it is on and 0 elsewhere), and the expectation of .
Now, suppose there are independent points uniformly chosen on the same disk, and let be the minimum distance from the center to the points, i.e. where are defined like the previous , for each of the points.
Then we have , from which we can get the density of the distribution of : it is on .