  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.
