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Cutting convex curves

We show that for any two convex curves $C_1$ and $C_2$ in $\mathbb R^d$ parametrized by $[0,1]$ with opposite orientations, there exists a hyperplane $H$ with the following property: For any $t\in [0,1]$ the points $C_1(t)$ and $C_2(t)$ are never in the same open halfspace bounded by $H$. This will be deduced from a more general result on equipartitions of ordered point sets by hyperplanes.