Monotonicity for the side lengths of stars inscribed in regular polygons

Question

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

Answer 1

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.

Answer 2

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