First but not least, I am not interested in an algorithm for finding the smallest circle enclosing N points in the plane!
Given a normal distribution of N points on a plane, centered on origo, and with the same standard deviation vertically and horizontally, can the expected radius of the smallest enclosing circle then be described by a closed expression?
My interest in this problem was caused by my interest in precision air pistol shooting. Given my education and nature, I immediately tried to understand and measure my (lack of) progress.
The traditional way of quantifying how well a set of N shots are grouped, is to measure the radius of the smallest circle enclosing the holes in the target. This makes sense if you are shooting on paper, and the group is tight, for the simple reason that later bullets could punch away the evidence of earlier ones. In addition to that, it is simple.
To me, however, it seems a waste to base the result on just the 2, 3 or so shots touching the smallest enclosing circle. I realised that the radius of a circle enclosing N shots would be a stochastic variable with a significantly larger variance than, say, the standard deviation. Today, when many target shooting clubs have electronic targets, capable of registering the exact coordinates for each hit, it would be natural to abandon the radius of the smallest enclosing circle, in favour of the standard deviation.
So, in hindsight, I should have asked for the variance of the radius of the smallest enclosing circle, not the expected value, since it is the smaller variance of the standard deviation which makes it superior to the smallest enclosing circle.
I am asking for a closed for expression, since a brute force simulation is less fun.
Hm. So maybe I should post this question in "Applied Mathematics" instead of "Mathematics".