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.

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!