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To start, think of a regular n-gon inscribed in a circle. If the vertices of the n-gon are all connected by drawing cords between the other vertices, then another smaller n-gon is created at the center of the circle, the "zero cell."

The zero cell contains the center of the circle, and thus by definition, it is unique.

What happens if instead of being evenly spaced, the n points are "randomly" selected from the circumference of the circle? On average, how many sides are to be expected? What would be the distribution of the number of sides of the zero cell?

Edited for clarity about how zero cell is constructed.

Edited again to specify that zero cell contains the center of the circle.

MaxW
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  • "another smaller n-gon is created at the center of the circle" is a little ambiguous. It's hard to answer if you do not say how the "zero cell" is constructed. – Lucas Jun 07 '13 at 02:53
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    draw chords between each vertex of the n-gon to every other vertex of the n-gon. I'll edit question for clarity... – MaxW Jun 07 '13 at 03:54
  • Does the zero cell only defined when it contains the centre of the circle? - any polygon formed points lying only in one half will not contain the centre. In other words, there will be lots of polygons formed by all those chords, how are you deciding which one to talk about? The one over the centre? The one made by chords with the smallest angles? etc. – Lucas Jun 07 '13 at 16:08
  • Yes the zero cell must contain the center of the circle. – MaxW Jun 07 '13 at 20:08
  • Well, that's quite a difficult question ;) – Lucas Jun 07 '13 at 22:50
  • Even a numerical answer is a bit of a mission. – Lucas Jun 07 '13 at 22:52
  • Do you need $n$ odd and greater than $4$? If $n$ is even then the example of a regular polygon has all points meeting in the center. For illustration, see examples in https://en.wikipedia.org/wiki/Complete_graph My intuition is for small perturbations away from regularity, even will generically be a triangle while odd will be an $n$-gon. I don't have intuition for how to approach the generic case (or even have the "zero cell" well defined -- it will not even be interior to the polygon if all the vertices are contained in one half of the boundary circle). – Neal Mar 09 '17 at 03:04
  • @Neal - Obviously if the vertices are for a even numbered regular n-gon then a n-gon will contain the center if n is odd and the center will be a vertex if n is even. That is why I specified "random points" around the circle. I'd guess that it would be some sort of skewed distribution where the most probably number of sides increases as the number of points increases. I'd expect the increase to be very gradual. So three sides for 10 points and maybe 4 for 500 points. Also I'd expect sides to drop fast. So a 20-gon would be high near impossible. – MaxW Mar 09 '17 at 04:24

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