Let $K\subset \mathbb{R}^n$ be a compact set and for each $r>0$, I would like to construct a partition of unity $\sum_{i=1}^N \xi_i=1$ of the set $K$ with the following properties: there exists a constant $C$, depending only on $n$ (but not $r$), such that
- the support of $\xi_i$ is contained in a ball with radius $r$,
- $|D\xi_i|\le C r^{-1}$,
- for each $x\in K$, the number of functions $(\xi_i)_i$ satisfying $\xi_i(x)\not=0$ is bounded by $C$.
My attempt:
For each $r>0$ and $x\in K$, my idea is to consider the test function such that $\eta_x=1$ on $B(x,r/2)$ and $\eta_x=0$ outside $r$ with the desired derivative bound. Then I try to find a sub-cover from $\bigcup_{x\in K}B(x,r/2)$, say $\bigcup_{i=1,\ldots, N(r,n)}B(x_i,r/2)$. Then I define $$\xi_i=\frac{\eta_{x_i}}{\sum_{k=1,\ldots, N(r,n)}\eta_{x_k}}, \quad i=1,\ldots, N(r,n).$$ The denominator is positive everywhere in $K$ since $\bigcup_{i=1,\ldots, N(r,n)}B(x_i,r/2)$ covers $K$.
The remaining point is to show Item 3, which reduces to the following question:
Let $r>0$, we consider the family of open balls $\bigcup_{x\in K}B(x,r/2)$ covering $K$. Since $K$ is compact, we know for each $r>0$, there exists a finite sub-cover, i.e., $N(r,n)\in \mathbb{N}$, $(x_i)_{i=1}^{N(r,n)}\subset K$ such that $K\subset \bigcup_{i=1,\ldots, N(r.n)}B(x_i,r/2)$. Hence for each $x\in K$, $x$ must lie in finite many such balls, say $M(x,r,n)$. It is clear that $M(x,r,n)\le N(r,n)$. May I know whether we can find a upper bound of the number $M(x,r,n)$, which is independent of $x$ and $r$?
The fact that the covering dimension of $\Bbb R^n$ equals $n$ (and so that of $K$ is at most $n$ too) we know that every open cover $\mathcal{U}$ of $K$ has an open refinement $\mathcal{V}$ such that each point of $K$ is at most $n+1$ many members of $\mathcal{V}$. I think it follows $n+1$ (or $\dim(K)+1$ really) will do.