Is the hemisphere a minimal isometric filling of its boundary circle, or can it be replaced by a Riemannian surface of smaller area without reducing the distance between any boundary points? Gromov posed the question and proved the strict minimality of the Euclidean hemisphere among surfaces homeomorphic to a disk. Ivanov considered more general Finsler metrics and proved that the Euclidean hemisphere is still minimal among disks, but there are many other Finsler disks that isometrically fill the circle and have the same area. In this talk I will present a discrete version of the problem:Can a cycle graph of length 2n be filled isometrically with a square-celled combinatorial surface made of less than n(n-1)/2 cells? (The filling is said isometric if the distance between each pair of boundary vertices, measured along the 1-skeleton graph of the filling surface, is not smaller than the distance along the boundary cycle.) This discrete question is equivalent to the continuous problem for self-reverse Finsler metrics, and is related to some known problems and structures including pseudo-line arrangements, minimizing the number of crossings between curves on surfaces, discrete differential forms, posets (including permutations with the Bruhat order), integral polygons, CAT(0) cubical complexes, integer linear programming, electrical networks and plabic graphs.
