Efficient cut for a subset of prescribed area

Wacharin Wichiramala

Abstract

We discuss an interesting isoperimetric problem on the plane. Given a set$S \subset R^2$ of finite area A and a real number 0 < a < A, we conjecture that there exists a set E such that S \ E has connected components of area a and length(E) is less than or equal to the shortest length needed to enclose a component of area a inside a disk of area A. When S is convex with infinite area, such E may have length at most $\sqrt{2\pi a}$ We prove this for the case that $a = \frac{A}{2}$ and there is a point inwhich any line through equally bisects the area of S. Moreover, E can be chosen so that it is on a straight line. nearring.