Cutting circles into equal area pieces (Q2752519)

The authors discuss the following results about dissections of the unit circle \(C\): NEWLINENEWLINENEWLINE(1) Any curve that cuts \(C\) into two pieces of equal area has length \(\geq 2\). Any diameter of \(C\) is an optimal cut (i.e. a cut of minimal length). NEWLINENEWLINENEWLINE(2) Cutting \(C\) into \(k\) pieces of equal area by straight lines that do not intersect in the interior of \(C\) is done by choosing some point \(c\) on the boundary of \(S\) and distributing \(k\) lines through \(c\) appropriately. This also gives an optimal solution of the problem, i.e. the sum of the lengths of the \(k\) lines is minimal. Every optimal solution can be transformed (by moving the cuts) to the solution described. NEWLINENEWLINENEWLINE(3) If \(C\) is cut into \(k\) pieces of equal area by straight lines such that these lines intersect in at most one Steiner point in \(C\), then the sum of the lengths of the lines is \(\geq k\). The optimal cut is realized in case the Steiner point is the center of \(C\). NEWLINENEWLINENEWLINEAll proofs only use elementary geometric arguments.