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Imagine I perform the following procedure:

[1] At time point $t_1$, I place a single point on a two-dimensional plane at the coordinate $(x, y) = (0, 0)$.

[2] At time point $t_2$, I center a circle of radius $r$ at the point placed during time $t_1$, and place another point somewhere within (or along the contour of) this circle with uniform probability across its area.

[3] For time point $t_i$, I first randomly select a previously placed point, with uniform probability for all points, and then I center a circle of radius $r$ at the point and place another point somewhere within (or along the contour of) this circle with uniform probability across its area.

GOTO [3] for $(N - 2)$ iterations.


Call $(p_1, ..., p_N) \in P$ the set of points $N$ placed on the surface via the above method.

My question is the following: What is the probability distribution for the radius and center position of the minimum circle that circumscribes (or covers) the set of $N$ points in $P$? How does this relate to the barycenter of the coordinates for the points in $P$ and the initial point at the origin?

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1 Answer 1

Of course the expected value of the center of gravity of the points is $(0,0)$. In one random simulation with $r=1$, for $n=10000$ points, I found the c.g. was $(0.085,-0.299)$. Although I cannot describe the distribution precisely mathematically, it is just what one would expect: each generated point is something like an $r$-fat point, and the statistics are consistent with that $r$-spread in variance. Here is a histogram for the same $n=10000$ points that yielded the near-zero c.g. above:
           Histogram
The minimum enclosing circle should also have an expected center of $(0,0)$, but I could not specify its variance without considerable analytical calculation.

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