Consider the collection $\mathcal{X}$ of 

* compact and connected subsets of $\mathbb{R}^2$,
* that are the closure of some open subset of $\mathbb{R}^2$, and
* have area $a$.

For any $\Omega \in \mathcal{X}$ and any $x,y \in \Omega$, let $d(x,y,\Omega)$ be the length of the shortest path lying entirely inside $\Omega$ and connecting $x$ to $y$.

Also, let $\mathbb{1}(x,y,\Omega)$ be the indicator function that equals $1$ if $x$ and $y$ are both in $\Omega$, and zero otherwise.

I am interested in the following problem:

> 
> $$\min\_{\Omega \in \mathcal{X}} \int\_{x\in \mathbb{R}^2}\int\_{y\in \mathbb{R}^2} d(x,y,\Omega) \mathbb{1}(x,y,\Omega) dx dy$$
> 
> 
> 

1. Is there a well-known solution to this problem?
2. Is it possible to show that a solution to this problem must be convex?
3. Is the problem even well-defined (e.g., does $d(x,y,\Omega)$ always exist? I tried to help guaranteeing it by including only closed and connected sets in $\mathcal{X}$, but I am not 100% sure that's enough)?
4. Is the problem guaranteed to have a solution without more regularity conditions?

Some related problems are described here: <https://link-springer-com.dist.lib.usu.edu/content/pdf/10.1007%2Fs10958-012-0717-3.pdf>. Unfortunately, these related problems either (1) focus on "networks" rather than "thick" sets, (2) focus right away on convex sets, or (3) do not fix the area and instead impose an additional "cost" to the minimization problem for increasing the area.