Let $P$ denote the perimeter function. It's not hard to prove that for any rectangle $R$ in $\mathbb{R}^2$, $R$ can be partitioned into a countable collection of squares $\{Q_k\}_{k=1}^{\infty}$ such that $R = \cup_kQ_k$, $\mathring{Q_k} \cap \mathring{Q_l}=\emptyset$, and $$\Sigma_{k=1}^{\infty}P(Q_k)\le 6P(R)$$
Now instead consider a convex polygon $K$ in $\mathbb{R}^2$, is it possible to find a universal constant $C$ and a partition $\{Q_k\}_{k=1}^{\infty}$ such that $R = \cup_kQ_k$, $\mathring{Q_k} \cap \mathring{Q_l}=\emptyset$, and $$\Sigma_{k=1}^{\infty}P(Q_k)\le CP(K)?$$
Also, what is the optimal value for $C$?
How about cases in $\mathbb{R}^n$? I could not even give a proof for $n$-dimensional rectangles.
Motivation of these questions:
It's a long story. I'm working on a variational problem related to sets of finite perimeter. A set of finite perimeter can be approximated by open sets with polyhedron boundary. A polyhedron can be divided into countable disjoint union of components. Note the perimeter of each component would not increase by considering its convex hull instead, thus it suffices to consider convex polygon. It is nice to use cubes to approximate sets of finite perimeter without increasing the perimeter too much.
I proved some covering lemmas to successfully give some characterizations of some function spaces with respect to sets of finite perimeter, and constructed a counterexample to show some inclusion relationships between those function spaces. I find combinatorics and some basic geometry knowledge is very helpful in my study.
These questions I asked here are not quite related to the problem I'm working on, since the covering lemma instead of "partition lemma" is enough. However, I'm just curious about the sharp case: why not give a partition lemma? And what is the optimal constant?
I'm not sure whether these results are known. Maybe they are very easy questions, since I googled them but didn't find anything.
Can anyone give me some references? Thanks in advance!