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Polygons which closest vertex is not neighbor

Source: 2019 Baltic Way P15

November 18, 2019
geometrycombinatorial geometry

Problem Statement

Let n≥4n \geq 4, and consider a (not necessarily convex) polygon P_1P_2\hdots P_n in the plane. Suppose that, for each PkP_k, there is a unique vertex Qk≠PkQ_k\ne P_k among P_1,\hdots, P_n that lies closest to it. The polygon is then said to be hostile if Qk≠Pk±1Q_k\ne P_{k\pm 1} for all kk (where P0=PnP_0 = P_n, Pn+1=P1P_{n+1} = P_1).
(a) Prove that no hostile polygon is convex. (b) Find all n≥4n \geq 4 for which there exists a hostile nn-gon.
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