  A theory is presented for the behavior of an array of multi-lamellar vesicles
(the onion phase) upon addition of solvent. A unique feature of this system is
the possibility to sustain pressure gradients by tension in the lamellae.
Tension enables the onions to remain stable beyond the unbinding point of a
flat lamellar stack. The model accounts for various concentration profiles and
interfaces developing in the onion as it swells. In particular, densely packed
`onion cores' are shown to appear, as observed in experiments. The formation of
interfaces and onion cores may represent an unusual example of stabilization of
curved interfaces in confined geometry.
