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

TL;DR

This paper investigates the minimal length of a split line dividing a regular polygon into equal parts, providing exact values for specific cases and asymptotic bounds for large m.

Contribution

It derives exact formulas for the minimal split line length in certain cases and establishes asymptotic bounds for large numbers of divisions.

Findings

Exact value of ${l_{2,3}}$ as $ extstyle rac{ ext{sqrt}(rac{ ext{sqrt}3 ext{pi}}{12})}$

Exact value of ${l_{3,3}}$ as $ extstyle rac{ ext{sqrt}3}{2}$

Asymptotic bounds for ${l_{m,n}}/ ext{sqrt}(m)$ as m approaches infinity.

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 {\rm{ctan}}\frac{\pi }{n}} \le \mathop {\lim }\limits_{m \to \infty } \frac{{{l_{m,n}}}}{{\sqrt m }} \le \sqrt {\frac{{\sqrt 3 }}{2}n{\rm{ctan}}\frac{\pi }{{\rm{n}}}} $