  We study the setting of committee elections, where a group of individuals
needs to collectively select a given size subset of available objects. This
model is relevant for a number of real-life scenarios including political
elections, participatory budgeting, and facility-location. We focus on the core
-- the classic notion of proportionality, stability and fairness. We show that
for a number of restricted domains including voter-interval,
candidate-interval, single-peaked, and single-crossing preferences the core is
non-empty and can be found in polynomial time. We show that the core might be
empty for strict top-monotonic preferences, yet we introduce a relaxation of
this class, which guarantees non-emptiness of the core. Our algorithms work
both in the randomized and discrete models. We also show that the classic known
proportional rules do not return committees from the core even for the most
restrictive domains among those we consider (in particular for 1D-Euclidean
preferences). We additionally prove a number of structural results that give
better insights into the nature of some of the restricted domains, and which in
particular give a better intuitive understanding of the class of top-monotonic
preferences.
