Equipping Lipschitz Boundaries with Riemannian Metric

Question

Suppose I have a bounded set $\Omega$ with a Lipschitz boundary. Is it possible to equip this boundary with a Riemannian distance function $d$ that satisfies: $\lambda |x-y| \leq d(x,y) \leq \mu|x-y|$ for all points in $\partial \Omega$? If not, what is the smallest regularity of the boundary one needs for this property to hold?

Edit: Ultimately what I want to do is the following. I have a set $\Omega \subset \mathbb R^d$ that is bounded, and $\partial \Omega$ is connected. Then, I define the following distance function, $d(x,y) = \inf ( \int_0^1 \|\gamma'\| dt \colon \gamma(0)=x, \gamma(1)=y \text{ and }\gamma \subset\partial \Omega )$. I am interested in conditions where there exist $\mu > 0$ such that $d(x,y) \leq \mu |x-y|$ for all points $x,y \in \partial \Omega$.

I suspect that this is true when $\partial \Omega$ is Lipschitz, in the sense that there exists a finite cover of $\partial \Omega$ where in each component of the cover, say $U_j$ it holds that $U_j \cap \partial \Omega$ is the graph of a Lipschitz function.

Answer 1

By compactness, there exists $r>0$ so that if points in $S=\partial \Omega$ are within Euclidean distance $\le r$, then they belong to an open subset $U\subset {\mathbb R}^n$ where $S$ is a graph of a Lipschitz function.

Let $f: D\to {\mathbb R}$ be an $L$-Lipschitz function defined on a domain in ${\mathbb R}^{n-1}$ with the graph $G_f$ equipped with the path-metric induced from ${\mathbb R}^n$. Then you define a $L'$-Lipschitz retraction $D\times {\mathbb R}\to G_f$ by $(x,t)\to (x,f(x))$. (Here $L'=\sqrt{L^2+1}$.) It follows that the path-metric on $G_f$ is bi-Lipschitz equivalent to the restriction of the distance function from ${\mathbb R}^n$: $$ |p-q|\le d(p,q)\le L |p-q|. $$ By compactness and connectivity of $S$, it follows that the diameter of $S$ (with the path-metric $d$) is finite (say, $D$). Therefore, if $p, q\in S$ satisfy $|p-q|>r$, then $$ |p-q|\le d(p,q)\le (D/r) |p-q| $$ so you have the required bi-Lipschitz inequality between the distances in this case. If $|p-q|<r$, then by the above observation you also have the required bi-Lipschitz inequality.