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This question is similar to the one here.

Now the question is, given a simple polygon with $Area>0$, regardless of whether it is convex or concave and with no opening, can we prove that the centroid of the polygon can never lie on the exact edge of the polygon?

If so, how?

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You should be able to do it with a crescent-shaped polygon. You shouldn't be able to do it for a convex polygon (basically because if the centroid of a convex polygon lies on an edge then all of the vertices of the polygon must lie on the corresponding line). –  Qiaochu Yuan May 4 '11 at 7:28
@Qiaochu, are you saying that a concave polygon-- if carefully constructed-- can have its centroid lies on the edge of the polygon? –  Graviton May 4 '11 at 8:37

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up vote 6 down vote accepted

The statement you suggest cannot be true, by continuity argument. If the polygon is "[" (imagine that the lines have non-zero thickness, i.e. "[" is a (right-angled) polygon), and if the horizontal edges are very short, then the centroid is inside, if they are very long then it is outside, so for some length it must be on the boundary (on the right vertical edge)

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