Sharp Reverse Isoperimetric Inequalities in Nonpositively Curved Cones

We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least 2 π have minimal area among all nonpositively curved disks of the same perimeter and the same total curvature; (2) geodesic triangles in a Euclidean (resp. hyperbolic) cone of angle at least 2 π have minimal area among all nonpositively curved geodesic triangles (resp. all geodesic triangles of curvature at most - 1) with the same side lengths and angles.