Before revision:  We show that the \gamma-meta-submodular families include well-known function classes including meta-submodular functions (\gamma=0), proportionally submodular (\gamma= 1 ), and diversity functions based on negative-type distances or Jensen-Shannon divergence (both \gamma=2) and (semi-)metric diversity)-approximation, independent of n.
After revision:  We show that the \gamma-meta-submodular families include well-known function classes including meta-submodular functions (\gamma=0), proportionally submodular (\gamma= 1 /3^{2\sigma)-approximation, independent of n.