The carbon-fluorine antibonding (sigma*) orbitals in a fluorocarbon cage are directed toward a central, common point. If the cage is not too large or too small, then the sigma* orbitals will overlap at that point. An added electron can occupy the resulting molecular orbital, suggesting that cage perfluorocarbons will have large electron affinities. This prediction is supported by electronic structure calculations of all of the fluorinated derivatives of tetrahedrane, cyclopropane, and cubane and of some other fluorinated cage and ring compounds. Perfluorododecahedrane (C20F20) is predicted to have an electron affinity of about 3.4 eV, which is equal to that of the fluorine atom. A few speculative extensions and applications are suggested.