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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)

Dummy View - NOT TO BE DELETED