It is shown that preferences which are continuous, convex and uniformly proper [Mas-Colell (1983)] on the positive cone of a Banach lattice can be represented by a quasi-concave utility function which is defined on a larger domain with non-empty interior. This utility function may be chosen to be either upper or lower semi-continuous on its domain, and continuous at each point of the positive cone. Conversely, any preference relation on the positive cone which is monotone and arises from such a utility function is shown to satisfy a condition which is slightly weaker than uniform properness but which (in the presence of appropriate compactness assumptions) is sufficient to establish the existence of quasi-equilibria. An example is presented to illuminate the role played by the uniformity requirement. © 1986.
Richard, S. F., & Zame, W. R. (1986). Proper preferences and quasi-concave utility functions. Journal of Mathematical Economics, 15(3), 231–247. https://doi.org/10.1016/0304-4068(86)90012-1