  This paper revisits the universal asymmetric $1 \to 2$ quantum cloning
problem. We identify the symmetry properties of this optimisation problem,
giving us access to the optimal quantum cloning map. Furthermore, we use the
bipolar theorem, a famous method from convex analysis, to completely
characterise the set of achievable single quantum clone qualities using the
fidelity as our figure of merit; from this it is easier to give the optimal
cloning map and to quantify the quality tradeoff in universal asymmetric
quantum cloning. Additionally, it allows us to analytically specify the set of
achievable single quantum clone qualities using a range of different figures of
merit.
