I am facing the following problem which I can not find a solution to, after several days of work :
First, define the following :
$\mu$ is locally bounded by $\omega$ if, for any subset $S$ of $\mathbb{R}^2$ of diameter at most $1$, $\mu(S) \leq \omega$. (this is the name I came up with, but if you know a more canonical name to this, please suggest it).
What I would like to know is the maximum (over all possible $\mu$) average value of $\mu(D)$ for $D \subset \mathbb{R}^2$ a disk of diameter $2$.
If it makes things easier, one might assume that the sets $S$ that are considered in the definition are disks too.
I'd be happy to give you more details if needed.
Thank you ! :)
Where does this problem come from ?
It comes from a graph coloring problem, and is a continuous generalization of a conjecture we have on finite graphs.
We work with unit disk graphs (graphs where nodes are disks of diameter $1$ and two disks are connected if they intersect), and would like to prove that the average degree of such a graph is always $\leq 4\omega$, where $\omega$ is the size of the biggest clique.
What bounds are currently known ?
With the Lebesgue measure, one gets a maximum value of $4w$.