  The core is a strong fairness notion in multiwinner voting and participatory
budgeting (PB). It is known that the core can be empty if we consider cardinal
utilities, but it is not known whether it is always satisfiable with
approval-ballots. In this short note, I show that in approval-based PB the core
can be empty for nearly all satisfaction functions that are based on the cost
of a project. In particular, I show that the core can be empty for the cost
satisfaction function, satisfaction functions based on diminishing marginal
returns and the share. However, it remains open whether the core can be empty
for the cardinality satisfaction function.
