Completeness of "weighted" shortest path metric

Question

I am trying to see when this type of metric is complete: Let $A$ be the set of $C^{1}$ paths in $U \in \mathbb{R}^{n}$. For any $x,y$ define $$\rho(x,y) = \inf_{\gamma \in A; \gamma(0) = x, \gamma(1) = y}\int_{0}^{1}V(\gamma(t))\|\gamma'(t)\|dt$$

We can assume that $U$ is convex, and that $V$ is as smooth as necessary.

What I've done so far:

• I was able to show that $\rho$ is a metric.
• Also, it seems that if one assumes $ c < V(x) < C $ for some $c,C > 0$ it can be shown to be strongly equivalent to a Euclidean norm.

However, I want to have some weaker constraints like $V > 0$. I also would appreciate any references for this.

Answer 1

Since you have put very little constraints on the subset $U$, there are so many different possibilities. I do not see how one can do anything other than list a few examples. I'll list two contrasting examples.

• From what you wrote, it follows that if $U$ is an open ball of finite radius and $c<V(x)<C$ for some $c,C>0$ then any straight line segment that terminates at a point of $\partial U$ contains a nonconvergent Cauchy sequence and so the metric is not complete. The same holds for any open, proper subset (whether or not convex).
• If $U \subset \mathbb{R}^2$ is the upper half plane and $V(x_1,x_2)=\frac{1}{x_2}$ then $U$ is the hyperbolic plane which is complete. More generally, if $U$ is any proper, open convex subset of the plane, or more generally any proper, open simply connected subset, then the uniformization theorem provides a function $V(x_1,x_2)$ which makes $U$ isometric to the hyperbolic plane, which is complete.