We study densities $\rho$ on the unit ball in
euclidean space which satisfy a Harnack type inequality and a volume growth condition for the measure
associated with $\rho$. For these densities a geometric theory can be developed which captures many features
of the theory of quasiconformal mappings. For example, we prove generalizations of the Gehring-Hayman
theorem, the radial limit theorem and find analogues of compression and expansion phenomena on the
1991 Mathematics Subject Classification: 30C65.