Consider $X$ to be a compact set in $[0,5]^2\subset \mathbb{R}^2$. Assume that $X$ is a union of translations of a unit ball. Then prove that $\partial X$ has a length bounded by $C$ where $C$ is independent of $X$.



Length of a boundary of union of unit balls in $\mathbb{R}^2$
1. Fix small $\epsilon>0$. We shall show that the union of balls with center in a given ball of radius $\epsilon$ must have bounded perimeter. Note each ball $D\_i$ has boundary expressible as a polar graph $$r=f\_i(\theta)$$ where $f\_i$ are uniformly bounded and uniformly Lipschitz. Then the boundary of $\cup\_iD\_i$ is given by the polar graph $$r=\sup\_if\_i(\theta)$$ which has bound and Lipschitz norm depending only on $\epsilon$, and hence length $L(\epsilon)$ depending only on $\epsilon$ (by the polar arc length formula from calculus).
2. Cover $[0,5]^2$ with balls $B\_1,
\dots, B\_{N(\epsilon)}$ of radius $\epsilon$. In each $B\_j$, let $X\_j$ be the the set of balls contributing to $X$ with center in $B\_j$. Then the length of $\partial X$ can be (not very sharply) bounded by $$\mathrm{length\,\, of\,\,} \partial X\le\sum\_{j=1}^{N(\epsilon)}\mathrm{length\,\, of\,\,} \partial X\_j\le N(\epsilon)L(\epsilon).$$


This can be optimized a bit by choosing the right $\epsilon$, but it is still an extremely weak bound as we are throwing away much of the boundary we compute.


