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

Anton Mosunov

Published 2020-02-11Version 1

In 2000 Bean and Laugesen proved that for every integer $n \geq 3$ the area bounded by the curve $$\prod\limits_{k = 1}^n\left|x\sin\left(\frac{k\pi}{n}\right) -y\cos\left(\frac{k\pi}{n}\right)\right| = 1 $$ 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.