The Japanese Theorem for Nonconvex Polygons - Limiting Behavior

Limiting Behavior

We have seen that the total inradii of regular polygons tend to the diameter of the circle. We now prove a slightly different theorem on the limiting behavior of cyclic polygons.

Theorem. Let \(A = \{ a_k : k \in {\mathbb Z}^+ , a_i \neq a_j \) for \( i \neq j\} \)   be a dense subset of a circle of radius \(R\) and let \(P(n)\) be the convex polygon formed by the vertices \( \{a_k : 1 \leq k \leq n \} . \) Then the sequence \( \left( r_{P(n)} \right)_{n=3}^{\infty} \) is eventually increasing and \( \lim_{n \rightarrow \infty} r_{P(n)} = 2R . \)

We leave the proof of the following lemma to the reader.

Lemma. \( \cos (x) + \cos (y) \geq \cos(x+y) + 1 \) for any \( (x,y) \) in the region bounded by \( x = 0, y = 0, \) and \( x + y = \pi , \) with equality only on the boundary.

Proof of theorem. Because \(A\) is dense, there is an \(M\)   such that the angle measures for every side of \(P(M)\) is less than \( \pi / 4 .\) Without loss of generality we may represent \(P(n+1) \) and \(P(n) \) as \( (\theta_1, \ldots, \theta_{n+1}) \) and \( ( \theta_1, \ldots, \theta_{n-1}, \theta_n + \theta_{n+1}) ,\) respectively. So,

\[ r_{P(n+1)} = R \left( 2 - (n+1) + \sum_{k=1}^{n+1} \cos \left( \frac{\theta_k}{2} \right) \right) \]

and

\[ r_{P(n)} = R \left( 2 - n + \sum_{k=1}^{n-1} \cos \left( \frac{\theta_k}{2} \right) + \cos \left( \frac{\theta_n + \theta_{n+1}}{2} \right) \right) . \]

From the preceding lemma it follows that \( r_{P(n)} < r_{P(n+1)} . \) So the sequence \( \left( r_{P(n)} \right)_{n=3}^{\infty} \) is eventually increasing.

By our theorem on regular polgyons we know that for a given number of vertices, the largest radial sum is obtained from the regular \(n\)-gon and this value is bounded above by \(2R .\) Consequently, the sequence \( \left( r_{P(n)} \right) \) is eventually increasing and is bounded above by \( 2R .\) Thus, to prove the theorem it suffices to show that for a given \( \epsilon > 0 , \) there exists an \(N\) such that \( r_{P(N)} > 2R - \epsilon .\) By the limiting theorem for regular polygons there exists an \( m > 4 \) such that \( r_{P_m} > 2R - \epsilon / 2 , \) where \( P_m \) is a regular \(m\)-gon. Since \(A\) is a dense set, there exist \(m\) distinct points \(A^{\prime} = \{ a_{n_1}, a_{n_2}, \ldots, a_{n_m} \} \subset A\) close enough to the \(m\) vertices of \( P_m \) that the polygon \( P^{\prime} \) with vertex set \( A^{\prime} \) has \( r_{P^{\prime}} > 2R - \epsilon \).∎