# Seek a partition of $\mathbb R^d$

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 $$F_k$$.

• I think your condition on $k$ is stated twice, no? In any case I suppose the two half-spaces $F_1=\{x:x_1\geq-c\}$ and $F_2=\{x:x_1\leq c\}$ should work. Feb 12 at 14:02
• (Or, as stated, $F_1=\mathbb R^d$...) Feb 12 at 14:05
• @PierrePC Many thanks for this quick example. Indeed, I look for the closed subsets $F_k$ of the form $F_k: = \{x : a\le |x|\le b\}$, while your construction still seems to work right? Feb 12 at 14:11
• The condition can be stated as "... and for every $x\in\mathbf{R}^d$, there exists $k$ such that $B(x,c)\subset F_k$".
– YCor
Feb 12 at 14:59
• But anyway the question is trivial as stated, so I assume that some requirements are missing. [Please make edits accordingly, rather than adding them in comments.]
– YCor
Feb 12 at 14:59