Let H=(N,E,w) be a hypergraph with a node set N={0,1,…,n-1}, a hyperedge set E⊆2N, and real edge-weights w(e) for e∈E. Given a convex n-gon P in the plane with vertices x0,x1,…,xn-1 which are arranged in this order clockwisely, let each node i∈N correspond to the vertex xi and define the area AP(H) of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For 0⩽i<j<k⩽n-1, a convex three-cut C(i,j,k) of N is {{i,…,j-1}, {j,…,k-1}, {k,…,n-1,0,…,i-1}} and its size cH(i,j,k) in H is defined as the sum of weights of edges e∈E such that e contains at least one node from each of {i,…,j-1}, {j,…,k-1} and {k,…,n-1,0,…,i-1}. We show that the following two conditions are equivalent:•AP(H)⩽AP(H′) for all convex n-gons P.•cH(i,j,k)⩽cH′(i,j,k) for all convex three-cuts C(i,j,k).From this property, a polynomial time algorithm for determining whether or not given weighted hypergraphs H and H′ satisfy “AP(H)⩽AP(H′) for all convex n-gons P” is immediately obtained.