Inside a public area $C$ (a polygon), there are several private land-plots $C_1,\dots,C_n$ (pairwise-disjoint simple polygons):
Currentlly, the public area that is outside the private land-plots (the set $C\setminus (C_1\cup\dots\cup C_n)$) is not connected - one cannot always walk from one part of the park to another without crossing a private plot.
The government wants to make the park path-connected by confiscating a tiny fraction of each of the private land-plots. Is this possible? I.e, is it possible, for every $\epsilon>0$, to remove a fraction $\epsilon$ of the area of each $C_i$, such that the set $C\setminus (C_1\cup\dots\cup C_n)$ becomes connected?
Intuitively, the government can "shave" a tiny fraction of the perimeter of each $C_i$, with width $\epsilon / Perimeter_i$. It seems "obvious" that the remainder will be path-connected. But how to prove this formally?