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 Ω ⊂ 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

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... 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 [L3] from W 1,p to BV .