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 $U\in \mathscr{U}$, the sub-collection $\mathscr{U}-\{U\}$ 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 $\|x-A\|:=\inf_{a\in A} \|x-a\|$.

Let $L>0$.  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
$$
\psi_U(x):= \frac{\|x-(K-U)\|^2}{\sum_{V \in \mathscr{V}}\,\|x-(K-V)\|^2}
$$
is $L$-Lipschitz?