We consider the model theoretic notion of convex orderability, which fits
strictly between the notions of VC-minimality and dp-minimality. In some classes
of algebraic theories, however, we show that convex orderability and
VC-minimality are equivalent, and use this to give a complete classification of
VC-minimal theories of ordered groups and abelian groups. Consequences for
fields are also considered, including a necessary condition for a theory of
valued fields to be quasi-VC-minimal. For example, the p-adics
are not quasi-VC-minimal.