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.