Given any compact, closed non-convex body $K \subset \mathbb{R}^n$ (containing NO holes!), is there a set of $N$ compact convex closed bodies, $\left\lbrace C\_i \right\rbrace\_{i=1}^N$ (where for every $i \in [N]$, $C\_i \subset \mathbb{R}^n$), such that $K = \cup\_{i=1}^N C\_i$ and $C\_i \subset K$ for every $i \in [N]$?

Note that $N$ can varies depending on $K$!

Please advise and thanks in advance.