Here is the claim:

Given a polytope $K$ in a unit ball in $\mathbb{R}^d$, there exists a universal constant $C(d)>0$ depending only on $d$ and a countable collection of convex polytopes $\{K_i\}_{i=1}^{\infty}$ such that $$K \subset \cup_{i=1}^{\infty} K_i$$and $$\sum_{i=1}^{\infty}S(K_i) \le C(d)S(K),$$where $S$ denotes the surface area.

Motivation:

One of the motivation of the claim is from the link I gave below. The other motivation is from my research. In my research I use again and again polytopes to approximate sets of finite perimeter (this is a term in geometric measure theory, basically means the finite surface area in a distributional sense), and in some cases, if I can use convex sets to approximate sets of finite perimeter, then it makes a difference, and some results can be refined.

In 2D the claim is straightforward, because the surface area is the perimeter, and taking a convex hull decreases the perimeter, thus the convex hull of $K$ is the natural cover, and $C(2)=1$.

In 3D, the argument in 2D is never true, see here the perimeter of a non-convex set.

I cannot prove or disprove this claim. I don't know the right tools to describe nonconvex polytope, because it can be composed of a lot of thick parts and thin parts, and their numbers cannot be controlled. The polytope can be as crazy as possible, but I guess there should be a way to cover it in a way as the claim says.

I haven't found any references dealing with this problem. I'm a beginner in convex geometry. Can anyone given me some references related to this problem? Thanks very much!