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