  We prove an $\Omega(n^2)$ lower bound on the query complexity of local
proportionality in the Robertson-Webb cake-cutting model. Local proportionality
requires that each agent prefer their allocation to the average of their
neighbors' allocations in some undirected social network. It is a weaker
fairness notion than envy-freeness, which also has query complexity
$\Omega(n^2)$, and generally incomparable to proportionality, which has query
complexity $\Theta(n \log n)$. This result separates the complexity of local
proportionality from that of ordinary proportionality, confirming the intuition
that finding a locally proportional allocation is a more difficult
computational problem.
