# The Japanese Theorem for Nonconvex Polygons - Limiting Behavior

Author(s):
David Richeson

### 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 $r_{P^{\prime}} > 2R - \epsilon$.∎

David Richeson, "The Japanese Theorem for Nonconvex Polygons - Limiting Behavior," Convergence (December 2013)