  Classic cake-cutting algorithms enable people with different preferences to
divide among them a heterogeneous resource (``cake''), such that the resulting
division is fair according to each agent's individual preferences. However,
these algorithms either ignore the geometry of the resource altogether, or
assume it is one-dimensional. In practice, it is often required to divide
multi-dimensional resources, such as land-estates or advertisement spaces in
print or electronic media. In such cases, the geometric shape of the allotted
piece is of crucial importance. For example, when building houses or designing
advertisements, in order to be useful, the allotments should be squares or
rectangles with bounded aspect-ratio. We thus introduce the problem of fair
land division --- fair division of a multi-dimensional resource wherein the
allocated piece must have a pre-specified geometric shape. We present
constructive division algorithms that satisfy the two most prominent fairness
criteria, namely envy-freeness and proportionality. In settings where
proportionality cannot be achieved due to the geometric constraints, our
algorithms provide a partially-proportional division, guaranteeing that the
fraction allocated to each agent be at least a certain positive constant. We
prove that in many natural settings the envy-freeness requirement is compatible
with the best attainable partial-proportionality.
