# Cutting a subset in many pieces with controlled perimeter

Let me premise that I am in no way an expert in the subject of this question.

Let's say we have a measure space $X$ with measure $m$ and a reasonable notion of perimeter for (nice enough) subsets of $X$. We denote $p(F)$ the perimeter of $F\subseteq X$.

I am interested in knowing classes of spaces for which the following property (P) holds:

(P) There is a constant $C<+\infty$ such that given any subset $F$ (nice enough) and any natural number $K$, we can split $F$ in disjoint pieces $F_1,\dots,F_K$ such that $m(F_j)\leq C \frac{m(F)}{K}$ and $p(F_j)\leq Cp(F)$ for any $j=1,\dots,K$.

Here nice enough may refer to having finite measure and perimeter and maybe some boundedness/compactness assumption.

Of course the question (if it is not stupid, because of some gross misunderstanding on my part) sounds absurdly vague, so let me point out that it seems to me that the property holds on euclidean spaces. If $F\subseteq\mathbb{R}^d$ one can cut $F$ using parallel hyperplanes in $K$ pieces having exactly $1/K$ of the total volume and everything should work with constant $C=1$.

I am especially interested in the graph case, i.e. $X=(V,E)$ is a (say countable and bounded-degree) graph with counting measure on vertices and perimeter of a subset $F$ measured by the number of edges connecting $F$ with its complement. Is $(P)$ a reasonable property? Are there interesting examples/counterexamples in the graph realm?

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I'll take the chance to mention the following nice paper of Aronov and Hubard. In it they answer a question of Nandakumar and Rao of whether a convex body can be partitioned into $n$ equal pieces of equal volume and equal perimeter. Their proof shows that this is always possible (in any dimension) when $n$ is prime. The prime condition is admittedly a bit strange and is likely just an artifact of the proof.