First, to show a limit exists: Since the sequence P_0, P_1, P_2, …

is bounded (all contained in P_0), compactness tells us there is a

convergent subsequence. But since the sequence is also monotone P_0 >

P_1 > … the original sequence must have the same limit as the

subsequence. Call this limiting set Q.

If the original polygon has k vertices, then so do all its successors

in the sequence. In the limit the number of vertices will stay the

same or decrease, but cannot increase. So Q is a polygon with at most

k vertices.

Applying the operation to the whole sequence just shifts it forward

one step, so the tail is the same, and Q must be invariant under the

operation. This is impossible unless Q is a single point. Moreover,

the operation preserves the mean of the (original) vertices, so Q must

be that point.

As all polygons have the same center of gravity, the limiting point is the centroid.

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