Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be a finite collection of open subsets covering $K$ satisfying the minimality property: *for every $U\in \mathcal{U}$, the sub-collection $\mathcal{U}-\{U\}$ does not cover $K$.*  

*Notation:* Given any subset $A\subseteq K$ denote the relative complement $K-A:=\{x\in K:\, x\not\in A\}$.  Given a point $x\in K$ and a subset $A\subseteq K$ let $\|x-A\|:=\inf_{a\in A} \|x-a\|$.

Is there a condition on the open cover $\mathcal{U}$ such that the partition of unity $\{\psi_U\}_{U \in \mathcal{U}}$ is such that each
$$
\psi_U(x):= \frac{\|x-(K-U)\|^2}{\sum_{V \in \mathcal{U}}\,\|x-(K-V)\|^2}
$$
is Lipschitz?  


If so, what are explicit bounds on the Lipschitz constant of the functions $\{\psi_U\}_{U\in \mathcal{U}}$?


*Gut-Feeling:* I expect that it is enough to assume that: there is some $\delta>0$ such that for every $x\in K$ there $\{u\in K:\, \|x-u\|<\delta\}$ is contained in some $U\in \mathcal{U}$.   However, I'm completely stumped...