UC Berkeley

F. Wehrung has asked: Given a family C of subsets of a set Ω, under what conditions will there exist a total ordering on Ω with respect to which every member of C is convex?¶ We look at the family P of subsets of Ω generated by C under certain partial operations which, when Ω is given with a total ordering, preserve convexity; we determine the possible structures of P under these operations if C, and hence P, is finite, and note a condition on that structure that is necessary and sufficient for there to exist an ordering of Ω of the desired sort. From this we obtain a criterion which works without the finiteness hypothesis on C.¶ Bounds are obtained on the cardinality of the set P generated under these operations by an n-element set C.¶ We end by noting some other ways of answering Wehrung’s question, using results in the literature. The bibliography lists still more related literature.