Inspired by the Happy Ending Problem

I recently came to learn of the Happy Ending Problem, thanks to Numberphile’s video. Apparently this was called the Happy Ending Problem because, in some way ,this problem has led to the marriage of George Szekeres and Esther Klein.

The theorem associate to this problem states that :
Any set of five points in the plane in general position (meaning that you can not have 3 points on a same line) has a subset of four points that form the vertices of a convex quadrilateral.

As I learned about this particular problem it lead me to Ramsey theory on which I hope to write a short post in a near future.

The happy ending problem and its generalisation inspired me the problem I wish to present here:

Let us start with a convex polygon with N sides. According to some rules (which are defined below) we add a point somewhere in the plane to create a new polygon which should remain convex. We then add another point and another and so on. We are interested in the convex shape obtained after adding an infinite number of points.

I will leave this problem here, and present in the future a part of the its solution.
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