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](https://i.stack.imgur.com/QGol3.png)


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](https://i.stack.imgur.com/WACuX.png)



Monotonicity for the side lengths of stars inscribed in regular polygons
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. 


