Suppose I have a box $D \subset \Bbb{R}^2$ (compact set). Denote $\mathcal{P}= \{ (\Omega_1,...,\Omega_n) : \bigcup_{i=1}^n \Omega_i = D,\ \Omega_i \cap \Omega_j =\emptyset\}$ the family of partitions of $D$ into open sets (the sets can overlap on their boundaries...)

Consider a functional $F$ defined on the subsets of $D$ such that $\lim_{|\Omega|\to 0}F(\Omega)=+\infty$. Suppose that there exists a partition $(\Omega_i)$ such that $\sum F(\Omega_i) < F(D)$ (so that the whole box is not optimal).

Then I consider the problem of finding the partition $(\Omega_i)$ which solves $$ \min_{(\Omega_i) \in \mathcal{P}}\sum_{i=1}^n F(\Omega_i).$$

The difference between this problem and the problems I've seen and worked until now is that here the number of members of the partition is not fixed.

The first condition tells us that $n$ is bounded above, since many parts means smaller volumes, which mean great $F(\Omega)$. The second condition says that $D$ is not the solution. In what I have in mind $F(\Omega)=G(\Omega)+\alpha \frac{Per(\Omega)}{|\Omega|}$ where $G \geq 0$ is something complicated.

And now the question...

  1. Do you know some references of articles which treat this kind of problems of optimal partitioning with variable number of parts?

  2. Do you know any available programs/toolboxes which are good for manipulating partitioning problems?


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