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

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}}}} $