  The ancient Gamow liquid drop model of nuclear energies has had a renewed
life as an interesting problem in the calculus of variations: Find a set
$\Omega \subset \mathbb R^3$ with given volume A that minimizes the sum of its
surface area and its Coulomb self energy. A ball minimizes the former and
maximizes the latter, but the conjecture is that a ball is always a minimizer
-- when there is a minimizer. Even the existence of minimizers for this
interesting geometric problem has not been shown in general. We prove the
existence of the absolute minimizer (over all $A$) of the energy divided by $A$
(the binding energy per particle). A second result of our work is a general
method for showing the existence of optimal sets in geometric minimization
problems, which we call the `method of the missing mass'. A third point is the
extension of the pulling back compactness lemma from $W^{1,p}$ to $BV$.
