Isoperimetric inequality for disconnected regions

The discrete isoperimetric inequality states that the regular n-gon has the largest area among all n-gons with a fixed perimeter p. In this paper, we extend the discrete isoperimetric inequality to disconnected regions in the hyperbolic plane, i.e., we permit the area to be divided between regions. We provide the necessary and sufficient conditions to ensure that the result holds for multiple polygons with areas that add up.