We model the observed black hole mass function under the assumption that black hole formation is controlled by the compactness of the stellar core at the time of collapse. Low-compactness stars are more likely to explode as supernovae and produce neutron stars, while high-compactness stars are more likely to be failed supernovae that produce black holes with the mass of the helium core of the star. Using three sequences of stellar models and marginalizing over a model for the completeness of the black hole mass function, we find that the compactness ... above which 50% of core collapses produce black holes is ... The models also predict that f = 0.18 (0.09 < f < 0.39) of core collapses fail. We tested four other criteria for black hole formation based on ... and ..., the compactnesses at enclosed masses of 2.0 or 3.0 rather than 2.5 M..., the mass of the iron core MFe, and the mass inside the oxygen burning shell MO. We found that ... works as well as ..., while ..., MFe and MO are significantly worse. As expected from the high compactness of 20-25 M... stars, black hole formation in this mass range provides a natural explanation of the red supergiant problem. (ProQuest: ... denotes formulae/symbols omitted.)