Concavifiable convex preference orderings are characterized and minimally concave utilities are constructed, using three different approaches. One involves the intersection of arbitrary lines with the three indifference surfaces, another involves conditions on the normals of two indifference surfaces and is related to the super-gradient map of a possible concave utility. In the third approach it is assumed that the ordering is induced by a twice-differentiable utility and Perror's integral of a certain expression formed from the derivatives is used. A possible economic interpretation of minimally concave utilities is suggested, and it is shown that one cannot select concave utilities so that they depend continuously on the ordering.
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*This research was started at the Center for Mathematical Studies in Economics and Management Science, where it was supported by National Science Foundation Grant GS 31346X. The early stages of the research were presented at the Mathematical Social Science Board Colloquium on Mathematical Economics in August 1974 at the University of California, Berkeley. The research was completed at the Weizmann Institute. The author wishes to thank all these institutions.