I have come across the following result regarding equitable division of a resource, which is a simple and immediate consequence of linear programming complementarity (in the infinite-dimensional case). I strongly suspect that it is a special case of some other well-known principle, perhaps in a related field such as economics or even geography, but as of yet I have not found any helpful leads. Can anyone suggest any results from which the statement below is a consequence? Any other leads will also be appreciated.

Let $R$ denote a contiguous resource (such as a piece of land) that is to be divided into $n$ pieces $R_1,\dots,R_n$ among $n$ agents (so agent $i$ gets piece $R_i$). Suppose that each agent has a continuous "utility density" function $u_i(x)>0$ defined on $R$ and that $R$ also has a continuous "population density" $f(x)>0$ defined on it. The total "utility" that agent $i$ receives is then $\iint_{R_i} f(x)u_i(x) dx$. One might consider the problem of choosing $R_1,\dots,R_n$ as "equitably" as possible, say by maximizing the minimum utility of the agents:

$\text{maximize}_{R_1,\dots,R_n}\lbrace\min_i \iint_{R_i} f(x)u_i(x) dx\rbrace$ subject to

$\bigcup_i R_i = R$ and $R_i \cap R_j = \emptyset \text{ for all }i\neq j$

It is easy to verify that, at the optimal solution $R_1^*,\dots,R_n^*$, all of the agents' utilities are equal.

Now, define $q_i^* := \iint_{R_i^*} f(x) dx$ (the population of the $i$th optimal piece), and consider the problem of choosing regions to maximize the *total* utilities of the agents, but under a logarithmic utility function, and imposing constraints on the populations in each piece:

$\text{maximize}_{R_1,\dots,R_n}\lbrace\sum_i \iint_{R_i} f(x)\log(u_i(x)) dx\rbrace$ subject to

$\iint_{R_i} f(x) dx = q_i^*$

$\bigcup_i R_i = R$ and $R_i \cap R_j = \emptyset \text{ for all }i\neq j$

It is not hard to show that the solution to this problem is the same as the solution to the original problem. I have uploaded a MATLAB script that demonstrates this principle using cvx at

http://menet.umn.edu/~jgc/mathoverflow.m

if anyone is interested.