Is the mean center of the vertices of a convex polygon always inside the polygon?

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.

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 $$t_1=...=t_n =\frac{1}{n}.$$