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I'm reposting this question from mathoverflow because it was off topic there; I hope I get an answer here.

As simple as that. I'm doing an R program where I need to order clockwise (or counter-clockwise) a bunch of points representing each a vertex of an irregular convex polygon and to do that I figured I could find a point inside, change to polar coordinates and calculate the angle for the vertices; but I don't know if the mean of the vertices always gives a point that lays inside the polygon.

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Yes, since a polygon $K$ is convex if for any natural number $n$ and any points $x_1, x_2 , ..x_n\in K$ and any numbers $0\leq t_1, t_2 , ..., t_n \leq 1$ such that $t_1 + t_2 +...t_n =1$ the point $$t_1 x_1 +t_2 x_2 +...t_n x_n $$ belong to $K.$ To answer your question it is enough to take vertices $x_1, x_2 , ..., x_k$ and $t_1=...=t_n =\frac{1}{n}.$

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  • $\begingroup$ Can you forward a proof of that statement? $\endgroup$ Oct 6 '20 at 16:00

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