The notion of an abstract convex geometry, due to Edelman and Jamison [7], offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara, Nakamura, and Okamoto [13] introduce the notion of a generalized convex shelling into R N and prove that a convex geometry may always be represented with such a shelling. We provide a new, shorter proof of their result using a representation theorem of [7] and deduce a different upper bound on the dimension of the shelling. Furthermore, in the spirit of Czédli [5], who shows that any 2-dimensional convex geometry may be embedded as circles in R 2 , we show that any convex geometry may be embedded as convex polygons in R 2 .

We propose a new abstract definition of equilibrium in the spirit of competitive equilibrium: a profile of alternatives and a public ordering (expressing prestige, price, or a social norm) such that each agent prefers his assigned alternative to all lower-ranked ones. The equilibrium operates in an abstract setting built upon a concept of convexity borrowed from convex geometry. We apply the concept to a variety of convex economies and relate it to Pareto optimality. The “magic” of linear equilibrium prices is put into perspective by establishing an analogy between linear functions in the standard convexity and “primitive orderings” in the abstract convexity. (JEL I11, I18, J44, K13)

Choice correspondence, Two-stage choice, Consideration sets, Axiomatization, D01,

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