A few results on the infimum of regular polygons equal-size split line

Abstract

If an n-side unit regular polygon is divided into m equal sized parts, then what is the minimum length of the split line $l_{m,n}$? This problem has its practical application in real world. This paper proved that $l_{2,3}=\sqrt{\frac{\sqrt{3}\pi }{12}}$, $l_{3,3}=\frac{\sqrt{3}}{2}$, and $\frac{1}{2}\sqrt{n\pi \mathrm{ctan}\frac{\pi }{n}}\le \underset{m\to \infty }{\lim }\frac{l_{m,n}}{\sqrt{m}}\le \sqrt{\frac{\sqrt{3}}{2}n\mathrm{ctan}\frac{\pi }{\mathrm{n}}}$

11 pages, 9 figures