Fair division of two-dimensional cake
The authors you cite have another paper on the topic.

Would you be satisfied with a model which assumes that the properties of the surfaces to be allocated can be summarized by an arbitrarily large but finite set of one-dimensional parameters (say length, depth, northmost point, eastmost point,... really as many as you want but finite) ? 

If this is satisfying to you, and you are ok with assuming that people have preferences over surfaces as described by the values of these parameters, you might find useful insights in the theory of fair allocation of bundles of multiple goods. A great (and free) introduction is "Fair Allocation Rules" by William Thomson. 

Of course when the dimensions represent parameters describing the shapes to be allocated, you are likely to have unusual preferences which are hard to work with and do not fit nicely with the existing results. Might be worth giving it a try though...