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.

  • $\begingroup$ "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. $\endgroup$ – Lucas Jun 7 '13 at 2:53
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    $\begingroup$ draw chords between each vertex of the n-gon to every other vertex of the n-gon. I'll edit question for clarity... $\endgroup$ – MaxW Jun 7 '13 at 3:54
  • $\begingroup$ 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. $\endgroup$ – Lucas Jun 7 '13 at 16:08
  • $\begingroup$ Yes the zero cell must contain the center of the circle. $\endgroup$ – MaxW Jun 7 '13 at 20:08
  • $\begingroup$ Well, that's quite a difficult question ;) $\endgroup$ – Lucas Jun 7 '13 at 22:50

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