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...

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 $\{\psi_U\}_{U\in \mathcal{U}}$?

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 $\|x-A\|:=\inf_{a\in A} \|x-a\|$.

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

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...