  We study the strong segregation limit for mixtures of Bose-Einstein
condensates modelled by a Gross-Pitaievskii functional. Our first main result
is that in presence of a trapping potential, for different intracomponent
strengths, the Thomas-Fermi limit is sufficient to determine the shape of the
minimizers. Our second main result is that for asymptotically equal
intracomponent strengths, one needs to go to the next order. The relevant limit
is a weighted isoperimetric problem. We then study the minimizers of this limit
problem, proving radial symmetry or symmetry breaking for different values of
the parameters. We finally show that in the absence of a confining potential,
even for non-equal intracomponent strengths, one needs to study a related
isoperimetric problem to gain information about the shape of the minimizers.
