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.