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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.

I particularly like the following real-world application (taken directly from their abstract).

Imagine that you are cooking chicken at a party. You will cut the raw chicken fillet with a sharp knife, marinate each of the pieces in a spicy sauce and then fry the pieces. The surface of each piece will be crispy and spicy. Can you cut the chicken so that all your guests get the same amount of crispy crust and the same amount of chicken?

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I hope their follow up paper (in differential equations, of course) addresses the variability of cooking times across all pieces of chicken (assuming a gas grill, but not assuming a spherical fillet). Gerhard "Please, No Choice Axiom Jokes" Paseman, 2013.06.10 – Gerhard Paseman Jun 10 '13 at 15:14
See also the MO question, "Cutting convex sets": – Joseph O'Rourke Jun 11 '13 at 10:43

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