We consider the free boundary isoperimetric problem in sectors of the Gauss plane. The solution is not always a circular arc as in sectors of the Euclidean plane. We prove that the solution is sometimes a ray and we conjecture that the solution is sometimes a "rounded n-gon" which we discovered computationally using Mathematica.
Adams, Elizabeth; Corwin, Ivan; Davis, Diana; Lee, Michelle; and Visocchi, Regina
"Isoperimetric Regions in Gauss Sectors,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 8:
1, Article 1.
Available at: https://scholar.rose-hulman.edu/rhumj/vol8/iss1/1