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Is it possible to prove that for all kinds of simple polygon, regardless of whether it is convex or concave and with no opening, the centroid of the polygon must ( or may not) lie inside the polygon?

The wiki link above gives example of polygon which has the centroid lying outside the polygon:

A non-convex object might have a centroid that is outside the figure itself. The centroid of a ring or a bowl, for example, lies in the object's central void.

But let's say my polygon has no opening, can it be proved that the centroid must lie inside the polygon?

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    $\begingroup$ Think of a boomerang. $\endgroup$
    – joriki
    Commented Mar 16, 2011 at 8:52
  • $\begingroup$ Alternatively, since you already understand the example of the polyon with an opening, imagine an arbitrarily narrow passage drilled to connect the opening to the outside and "simplify" the polygon. Since you can make the passage as narrow as you want, you can move the centroid as little as you want, so you can see it doesn't have to move inside the polygon. $\endgroup$
    – joriki
    Commented Mar 16, 2011 at 9:45

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Comment as answer as requested:

  • Think of a boomerang.
  • Alternatively, since you already understand the example of the polyon with an opening, imagine an arbitrarily narrow passage drilled to connect the opening to the outside and "simplify" the polygon. Since you can make the passage as narrow as you want, you can move the centroid as little as you want, so you can see it doesn't have to move inside the polygon.
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  • $\begingroup$ Or imagine a highjumper doing the [Fosbury Flop] (en.wikipedia.org/wiki/Fosbury_Flop): the body goes over the bar while the centre of mass goes under it. Now take a photograph (which removes time and a spatial dimension). $\endgroup$
    – Henry
    Commented Mar 16, 2011 at 10:26

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