Cutting convex curves

We show that for any two convex curves C1C1 and C2C2 in RdRd parametrized by [0,1][0,1] with opposite orientations, there exists a hyperplane HH with the following property: For any t∈[0,1]t∈[0,1] the points C1(t)C1(t) and C2(t)C2(t) are never in the same open half space bounded by HH. This will be deduced from a more general result on equipartitions of ordered point sets by hyperplanes.