 # The Japanese Theorem for Nonconvex Polygons - Extreme Values for Cyclic Polygons

Author(s):
David Richeson

### Extreme Values for Cyclic Polygons

We conclude by looking for extreme values of the total (signed) inradius for cyclic $n$-gons. To do so we must look closer at the space of cyclic $n$-gons inscribed in a circle of radius $R$, which we denote $\mathcal{P}_{R,n} = \mathcal{P}_n$, and the function $f : \mathcal{P}_n \rightarrow {\mathbb R}$, given by $f(P) = \tilde{r}_P .$

As we did for convex cyclic $n$-gons, we assume that $p_1 = (R,0)$ and we identify each polygon in $\mathcal{P}_n$ with the vector of central angles $(\theta_1, \ldots, \theta_n )$, but now the $\theta_i$ can take on any value; they can even be negative. So, perhaps the most simple representation is

$\mathcal{P}_n = \{ (\theta_1, \ldots, \theta_n) \in {\mathbb R}^n : \theta_1 + \cdots + \theta_n = 2 k \pi \text{ for some } k \in {\mathbb Z} \} ,$

but this representation hides the fact that different $n$-tuples can correspond to the same polygon. For instance, $(\theta_1, \ldots, \theta_{n-1}) \text{ and } (\theta_1, \ldots, \theta_{n-1}, 2 \pi + \theta_n )$ represent the same polygon. Specifically, we have an equivalence relation $\sim$ in which $(\theta_1, \ldots, \theta_n) \sim (\theta_1^{\prime}, \ldots, \theta_n^{\prime})$ provided $(\theta_1, \ldots, \theta_n) - (\theta_1^{\prime}, \ldots, \theta_n^{\prime}) = (k_1 2 \pi, \ldots, k_n 2 \pi)$ for some $k_1, \ldots, k_n \in {\mathbb Z} .$ So

$\mathcal{P}_n = \{ (\theta_1, \ldots, \theta_n) \in {\mathbb R}^n : \theta_1 + \cdots + \theta_n = 2 k \pi \text{ for some } k \in {\mathbb Z} \} / \sim .$

Let us simplify this even more. First of all, we may assume that the angles are between $0$ and $2 \pi .$ Then we only have ambiguity at the endpoints. Second, we observe that the $n$th coordinate is superfluous since it is uniquely determined by the first $n-1$ coordinates. So,

$\mathcal{P}_n = \{ (\theta_1, \ldots, \theta_{n-1}) : 0 \leq \theta_i \leq 2 \pi \} / \sim$

$= [0, 2 \pi]^{n-1}/ \sim .$

This last representation gives us the best way to view $\mathcal{P}_n .$ The space is the $(n-1)$-dimensional cube $[0, 2 \pi]^{n-1}$ with the opposite faces glued together. In other words, $\mathcal{P}_n$ is the $(n-1)$-dimensional torus. Another way of seeing that this is the topology of $\mathcal{P}_n$ is to notice that there is a circle of possible values for each of the first  $n - 1$ angles $\theta_i$. So

$\mathcal{P}_n = \underbrace{ S^1 \times \cdots \times S^1}_{n-1} ,$

where $S^1$ is a circle. This is a standard way of representing the $(n-1)$-torus.

Earlier we gave the following explicit expression for the radial sum function $f : \mathcal{P}_n^c \rightarrow {\mathbb R} ,$ $f ( \theta_1, \ldots, \theta_n ) = R \left( 2 - n + \sum_{i=1}^n \cos \left(\frac{\theta_i}{2} \right) \right) .$

We can use an identical argument, but now using the generalized Carnot's theorem, to obtain an expression for $f : \mathcal{P}_n \rightarrow {\mathbb R} .$ Let $P = ( \theta_1, \ldots, \theta_n) \in \mathcal{P}_n .$ Specifically, suppose $\theta_i \in [0, 2 \pi)$ for all $i, \theta_1 + \cdots + \theta_n = 2 k \pi \text{ for some } k \in {\mathbb Z}$, and $p$ and $q$ are the numbers of positively and negatively oriented triangles in some triangulation of $P$, then

$f(P) = f ( \theta_1, \ldots, \theta_n ) = R \left( q - p + \sum_{i=1}^n \cos \left( \frac{\theta_i}{2} \right) \right) .$

Finally, as before, we may use this function to determine the locations of the extreme values.

Theorem. Consider the function $f : \mathcal{P}_n \rightarrow {\mathbb R}$ given by $f(P) = \tilde{r}_P .$

1. The unique absolute maximum of $f$ is the regular $n$-gon with vertices situated counterclockwise.
2. The unique absolute minimum of $f$ is the regular $n$-gon with vertices situated clockwise.
3. The function $f$ has no relative, non-absolute extrema.

We omit the proof of this theorem. It is similar to, but, because of the presence of $q$ and $p$ in the expression for $f$, slightly more subtle than the proof of the corresponding theorem for convex cyclic polygons.

David Richeson, "The Japanese Theorem for Nonconvex Polygons - Extreme Values for Cyclic Polygons," Convergence (December 2013)

## Dummy View - NOT TO BE DELETED

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