  We present a general approach for calculating densities and other local
quantities of trapped Fermi gases, when the cloud shape is distorted with
respect to the trap shape due to global energy considerations. Our approach
provides a consistent way to explore physics beyond the local density
approximation, if this is necessary due to the distortion. We illustrate this
by analyzing in detail experimentally observed distortions in an imbalanced
Fermi mixture in an elongated trap. In particular, we demonstrate in that case
dramatic deviations from ellipsoidal cloud shapes arising from the competition
between surface and bulk energies.
