On the Area Bounded by the Curve $\prod_{k = 1}^n |x\sin(k\pi/n)-y\cos(k\pi/n)| = 1$

Abstract

For a positive integer $n$, let $$ F_n^*(X, Y) = \prod\limits_{k = 1}^n\left(X\sin\left(\frac{k\pi}{n}\right) -Y\cos\left(\frac{k\pi}{n}\right)\right). $$ In 2000 Bean and Laugesen proved that for every $n \geq 3$ the area bounded by the curve $|F_n^*(x, y)| = 1$ is equal to $4^{1 - 1/n}B\left(\frac{1}{2} - \frac{1}{n}, \frac{1}{2}\right)$, where $B(x, y)$ is the beta function. We provide an elementary proof of this fact based on the polar formula for the area calculation. We also prove that $$ F_n^*(X, Y) = 2^{1 - n}\sum\limits_{\substack{1 \leq k \leq n\\\text{$k$ is odd}}}(-1)^{\frac{k - 1}{2}}\binom{n}{k}X^{n - k}Y^k $$ and demonstrate that $\ell_n = 2^{n - 1 - \nu_2(n)}$ is the smallest positive integer such that the binary form $\ell_n F_n^*(X, Y)$ has integer coefficients. Here $\nu_2(n)$ denotes the $2$-adic order of $n$.