This article establishes versions of Moulin’s [On strategy-proofness and single peakedness, Public Choice 35 (1980), 31–38] characterizations of various classes of strategy-proof social choice functions when the domain con-sists of all profiles of single-peaked preferences on an arbitrary subset of the real line. Two results are established that show that the median of 2n + 1 numbers can be expressed using a combination of minimization and maxi-mization operations applied to subsets of these numbers when either these subsets or the numbers themselves are restricted in a particular way. These results are used to show how Moulin’s characterizations of generalized median social choice functions can be obtained as corollaries of his characterization of min-max social choice functions.