The following variant of the fair cake-cutting problem was introduced by Ted Hill in 1983.
This geographic constraint distinguishes this problem from classic fair cake-cutting.
There are cases in which the problem cannot be solved: In 1983, Hill proved that, if each single point in D has a value of 0 for all countries, and the boundary of D has a value of 0 for all countries, then there exists a proportional division with the adjacency constraint.
[1] 4 years later, Anatole Beck described a protocol for attaining such a division.
Find a Riemann mapping h that maps D to the unit disc, such that for all countries, the value of every circle centered at the origin is 0 and the value of every radius from the origin is 0 (the existence of such an h is proved by a counting argument).
Its value for the other countries is strictly less than 1/n, so we can give to Ci a small additional piece connecting it to its allocated disc.
The remainder is simply-connected and has a value of at least (n − 1)/n to the remaining n − 1 countries, so the division can proceed recursively in step 1.
Again there are two cases: If C1 is one of the countries bidding D2, then just give it D2 (which is slightly smaller than the original D1) and the connecting sector.
To solve this, C2 is allowed to take another sector, this time of length less than 1 so that it doesn't harm the connectivity.
If the territory D is k-connected with a finite k, then the division can proceed by induction on k. When k = 1, D is simply-connected and can be divided by the protocol of the previous section.
There is an uncountable infinity of pairwise-disjoint lines connecting B1 and Bk and contained in D. But the measure of D is finite, so the number of lines with a positive measure must be finite.