Putnam 2012, Problem B-2

Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P) > 0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P$, then $n > c(P)/V^2$.

Solution:
Fix a face $F$ of the polyhedron with area $A$. Suppose $F$ is completely covered by balls of radii $r_1, \ldots, r_n$ whose volumes sum to $V$. Then on one hand,
$$\sum_{i=1}^n \frac{4}{3}\pi r_i^3 = V.$$

On the other hand, the intersection of a ball of radius $r$ with the plane containing $F$ is a disc of radius at most $r$, which covers a piece of $F$ of area at most $\pi r^2$; therefore
$$\sum_{i=1}^n \pi r_i^2 \geq A.$$

By writing $n$ as $\sum_{i=1}^n 1$ and applying Hölder's inequality, we obtain
$$nV^2 \geq \left(\sum_{i=1}^n \left(\frac{4}{3}\pi r_i^3\right)^{2/3}\right)^3 \geq \frac{16}{9\pi}A^3.$$

Consequently, any value of $c(P)$ less than $\frac{16}{9\pi}A^3$ works.