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Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet).

enter image description here

I've been trying to find, in terms of $n$, bounds on the area of the largest cell, excluding the centre cell when $n$ is odd.

It seems that, for every value of $n$, the largest non-centre cell has roughly similar area as the outermost cell, whose area is $\frac{1}{4}\tan{\frac{\pi}{n}}\approx\frac{\pi}{4n}$.

enter image description here

That is, $(\text{area of largest non-centre cell})$ is on the order of $n^{-1}$.

(This claim has numerical evidence: the first graph in this answer shows that, in a regular $n$-gon of radius $1$ with diagonals, up to $n=200$, the radius of the largest disk that fits within a cell, is approximately $3n^{-1.5}$; scaling to a regular $n$-gon of side length $1$, the largest disk would have an area of approximately $\frac{9}{4\pi n}$.)

Question:

In a regular $n$-gon of side length $1$ with diagonals, what is the infimum and supremum of $n\times(\text{area of largest non-centre cell})$ ?

(The number of cells is approximately $\frac{1}{24}n^4$ for large $n$.)

EDIT: If we don't know the answer to this question, then can we at least show that $n\times(\text{area of largest non-centre cell})$ has an upper bound?

Cross-posted on MSE.

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    $\begingroup$ Here is a strategy for a reasonable upper bound: Don’t look at all the diagonals! Look only at the nearly-vertical and nearly-horizontal diagonals (i.e. diagonals from one vertex to the vertex with next-largest x- or y-coordinate); the area of the largest quadrilateral thus cutoff will be an upper bound for the largest area cut off by all the diagonals. $\endgroup$
    – user44143
    Commented Jan 7, 2023 at 4:24
  • $\begingroup$ The largest quadrilateral thus cutoff, excluding the centre quadrilateral, is at least as large as a quadrilateral next to the centre quadrilateral, which has an area of approximately $1/4$ for large $n$. But $n/4$ is unbounded. $\endgroup$
    – Dan
    Commented Jan 7, 2023 at 7:10
  • $\begingroup$ Can you prove at least that no cell except the central contains a circle of radius $10000/\sqrt{n}$? $\endgroup$
    – Fedor Petrov
    Commented Jan 7, 2023 at 8:30
  • $\begingroup$ @FedorPetrov I do not know how to prove that. $\endgroup$
    – Dan
    Commented Jan 7, 2023 at 11:21

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