Monotonicity for the side lengths of stars inscribed in regular polygons

Fix integers $l\ge 1$ and $n \ge 3$, and let $P_n$ denote the boundary of the regular $n$-sided polygon in the plane. We define a $(2l+1)$-pointed equilateral star to be a cyclically ordered list of points $\{v_0,v_1,\dots v_{2l}, v_{2l+1}=v_0\} \subseteq P_n$ such that the adjacent distances $\|v_i-v_{i+1}\|$ are all equal to some constant $r$ (called the side length of the star), and such that the path "winds" $l$ times around the center of $P_n$. For example, 3-pointed equilateral stars are exactly equilateral triangles, and 5-pointed equilateral stars agree with the intuitive image of star as depicted below. Observe also that "equilateral" is equivalent to "equiangular" for $l = 1$, but this does not necessarily hold for $l \ge 2$.

Some examples of stars inscribed in regular polygons

It turns out that so long as $n \ge 4l+2$, for any point $x\in P_n$ there exists a unique $(2l+1)$-pointed equilateral star inscribed in $P_n$ which contains $x$ as one of its vertices. So for $n \ge 4l+2$, we may define the side length function $s_{2l+1}\colon P_n \to \mathbb{R}$, which assigns to each point $x \in P_n$ the side length of the unique star containing $x$. We are able to prove that this side length function is continuous.

Now we are curious about the extrema of this side length function, for which the following is already known: If $x \in P_n$ is such that the star containing $x$ also contains a vertex of $P_n$, then the symmetry of $P_n$ gives that $s_{2l+1}$ has a local extremum at $x$; moreover, for any $x$ with this property, we must have that $s_{2l+1}(x)$ achieves the same value. Likewise, if $x \in P_n$ is such that the star containing $x$ also contains a midpoint of an edge of $P_n$, then $s_{2l+1}$ has a local extremum at $x$, and all points with this property must achieve the same value.

Let $n\ge 4l+2$. We conjecture (Conjecture 5.10 of https://arxiv.org/abs/1807.10971) that the only extrema of the side length function $s_{2l+1}\colon P_n \to \mathbb{R}$ are the extrema described above, where the side-length function achieves a maximum if the star containing $x$ contains a vertex of $P_n$, and where the side-length function achieves a minimum if the star containing $x$ contains a midpoint of an edge of $P_n$. We do not know how to show that there are no other local extrema. Any ideas for how to show this?

The result is proven in the above paper when $l=1$ or when $2l+1$ divides $n$. Computational results corroborate our conjecture in the general case, as in the following plots, which illustrate the monotonicity of the side-length function between (known and clearly visible) known local maxima and minima. The leftmost figure is for 3-stars inscribed in $P_6$, the middle figure is for 3-stars inscribed in $P_7$, and the rightmost figure is for 5-stars inscribed in $P_{11}$. The vertical axis is the side-length of the inscribed star, and the horizontal axis parametrizes the locations of the vertices of the star. A few small apparent non-global extrema in the middle and right plots are simply noise due to numerical rounding issues.

Plots of the side length function

Instead of considering the closed stars, it is convenient to consider inscribed broken lines with constant leg lengths.

Lemma. Let $AOB$ be an angle, let the points $X$ and $Y$ move along $AO$ and $OB$ monotonically, so that $X$ moves with the constant speed, and $XY$ is constant. Then the coordinate of $Y$ changes concavely.

Proof. A direct computation via cosines rule.

Corollary. Let $X_0X_1\dots X_{2\ell+1}$ be a broken line with constant leg length inscribef into a regular polygon, whise vertices move monotonically along the sides of the polygon, $X_0$ with a constant speed. Then $X_{2\ell+1}$ moves concavely, until some vertex of the broken line meets a vertex of the polygon.

Proof. A composition of increasing concave functions is also concave.

Now return back to the original question. Let $X_0X_1\dots X_{2\ell+1}$ be an inscribed star (whose vertices are distinct from vertices of the polygon and modpoints of its sides), $X_{2\ell+1}=X_0$. Let its points move as in the corollary ($X_0$ and $X_{2\ell+1}$ become distinct). Either forward or backward, this broken line meets a configuration symmetric to the original one (which is also closed) before tracing a vertex of the polygon. By concavity, between these two positions $X_{2\ell+1}$ went ahead'' of $X_0$, which means that the inscribed stars starting at those positions of $X_0$ were shorter.

Notice that, due to the symmetry, between our two positions there was exactly one with a midpoint of a polygon's side on the star. This yields the required monotonicity.

• Thanks for this insight! Seems like it does give us a clean answer to this problem. We found a (gmail) email address in one your arxiv papers, and we'll contact you offline through that--let us know if you don't get our email! – Adam Quinn Jaffe Aug 31 '18 at 1:06

In further detail, we are currently able to prove the above conjecture in two special cases:

First, suppose that $2l+1$ divides $n$. Then, it is straightforward to construct the star that contains $x$ for any $x \in P_n$, as follows: If we imagine $P_n$ as the boundary of the convex hull of all of the $n$th roots of unity, then an arbitrary point $x \in P_n$ can be written without loss of generality as $x = (1-t)\omega_n+t\omega_n$ for $t \in [0,1)$ and $\omega_n = \exp(2\pi i/n)$. Then, $\{x,x\omega_n^{l},x\omega_n^{2l},\dots \}$ is a star in $P_n$ that contains $x$, and we can compute that its side length is

$$|x|\cdot|1-\omega_n^{l}| = \sin\bigg(\frac{\pi l}{2l+1}\bigg)\sqrt{4\sin^2\bigg(\frac{\pi}{n}\bigg)t^2-4\sin^2\bigg(\frac{\pi}{n}\bigg)t+1}$$

By uniqueness, this must be the value of $s_{2l+1}(x)$. Since this is quadratic in $t \in [0,1)$, we can easily see that it achieves its maximum at $t = 0$, its minimum at $t = 1/2$, and that it has no other extrema, as desired.

Second, suppose that $l = 1$. Then for any $x \in P_n$, let $PQR$ be the unique equilateral triangle in $P_n$ that contains $x$. The vertices of $PQR$ must lie on three distinct edges of $P_n$, which can be extended to form triangle $ABC$. Then, we can construct a triangle $TUV$ which is both circumscribed about $ABC$ and parallel to $PQR$. This is illustrated in the following image:

An illustration of the above construction

We can then show that $RQ\cdot VU = k_{ABC}$, where $k_{ABC}$ is a constant depending only on $ABC$. Lastly, we show that $VU$ is proportional to the cosine of a certain angle, which establishes that $RQ$ is monotonic between its extrema.

We're interested in proving this result for general $(n,l)$, but it seems that neither method described above generalizes appropriately. Any ideas are welcome! For more details about any of the above, check out our preprint at https://arxiv.org/abs/1807.10971