Let $c>0$ be given. I look for $n\ge 1$ and a collection of closed subsets $(F_i: 1\le i\le n)$ such that

$$\bigcup_{1\le i\le n} {\rm int}(F_i)= \mathbb R^d,$$

and for every $x\in \mathbb R^d$, there exists $k\in P(x)$ satisfying $x\in {\rm int}(F_k)$ and

$$\inf \big\{|x-y|: y\in \partial F_k\big\}\ge c,$$

where $P(x):=\{1\le i\le n: x\in {\rm int}(F_i)\}$. Here ${\rm int}(F_k)$ denotes the interior of $F_k$ and $\partial F_k$ denotes the boundary of $F_k$.