# 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," Loci (December 2013)