We present a general approach for describing trapped Fermi gases, when the cloud shape is distorted
with respect to the trap shape. 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 a trapped imbalanced Fermi mixture. In particular, we demonstrate
in that case dramatic deviations from ellipsoidal cloud shapes arising from the competition between
surface and bulk energies
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