  Mixable loss functions are of fundamental importance in the context of
prediction with expert advice in the online setting since they characterize
fast learning rates. By re-interpreting properness from the point of view of
differential geometry, we provide a simple geometric characterization of
mixability for the binary and multi-class cases: a proper loss function $\ell$
is $\eta$-mixable if and only if the superpredition set $\textrm{spr}(\eta
\ell)$ of the scaled loss function $\eta \ell$ slides freely inside the
superprediction set $\textrm{spr}(\ell_{\log})$ of the log loss $\ell_{\log}$,
under fairly general assumptions on the differentiability of $\ell$. Our
approach provides a way to treat some concepts concerning loss functions (like
properness) in a ''coordinate-free'' manner and reconciles previous results
obtained for mixable loss functions for the binary and the multi-class cases.
