Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be finite open subsets covering $K$.  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
$$
\psi_U(x):= \frac{\|x-(K-U)\|^2}{\sum_{U \in \mathscr{V}}\,\|x-(K-V)\|^2}
$$
is $L$-Lipschitz?