  We generalize quasi-arithmetic means beyond scalars by considering the
gradient map of a Legendre type real-valued function. The gradient map of a
Legendre type function is proven strictly comonotone with a global inverse. It
thus yields a generalization of strictly mononotone and differentiable
functions generating scalar quasi-arithmetic means. Furthermore, the Legendre
transformation gives rise to pairs of dual quasi-arithmetic averages via the
convex duality. We study the invariance and equivariance properties under
affine transformations of quasi-arithmetic averages via the lens of dually flat
spaces of information geometry. We show how these quasi-arithmetic averages are
used to express points on dual geodesics and sided barycenters in the dual
affine coordinate systems. We then consider quasi-arithmetic mixtures and
describe several parametric and non-parametric statistical models which are
closed under the quasi-arithmetic mixture operation.
