Let $C$ be a hypercube in $\mathbb{R}^D$ with edge length of $L$. Let $\mathcal{P}_1,\ldots,\mathcal{P}_K$ be $K$ convex polytopes that partition $C$. Let $S_k$ be the surface area of the polytope $\mathcal{P}\_k$, where $k\in \{1,\ldots, K \}$. Define $S=\sum_{k=1}^K S_k$, what is the upper bound and lower bound of $S$? In particular, I'm interested in the bounds which can be expressed as function of $K,D$ and $L$.

In case it is not clear, the

*surface area*of a convex polytope $\mathcal{P}$ is the sum of $(D-1)$-dimensional Lebesgue measure of the facets of $\mathcal{P}$.There may have some nice results when $K\rightarrow\infty$ and reformulating this problem as a tessellation induced by some random processes. However, I'm interested in small $K$, say $1< K<50$.

I guess this problem has been solved, but I'm struggling to find good literature.