Two equivalent measures on weighted hypergraphs Original Research Article

Let image be a hypergraph with a node set image, a hyperedge set image, and real edge-weights image for image. Given a convex n-gon P in the plane with vertices image which are arranged in this order clockwisely, let each node image correspond to the vertex image and define the area image of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For image, a convex three-cut image of N is image, image, image and its size image in H is defined as the sum of weights of edges image such that e contains at least one node from each of image, image and image. We show that the following two conditions are equivalent: