Length of a boundary of union of unit balls in $\mathbb{R}^2$ 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 $X$.
 A: *

*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).

*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.
