Almgren's Three-Legged Starfish

In this note we use classical tools from min-max and hyperbolic geometry to substantiate a folklore example in Almgren–Pitts min-max theory, the three-legged starfish metric on a 2-sphere, whose systolic length, Almgren–Pitts width, and Gromov–Guth width are attained by figure-eight'' geodesics. We also recover a hyperbolic geometry fact about figure-eight’’ geodesics using min-max.