When can a majority of voters find common ground, that is, a position they all agree upon? How does the shape of the political spectrum influence the outcome? When mathematical objects have a social interpretation, the associated theorems have social applications. In this article we give examples of situations where sets model preferences and develop extensions of classical theorems about convex sets, such as Helly's theorem, that can be used in the analysis of voting in "agreeable" societies.

When is agreement possible? An important aspect of group decision-making is the question of how a group makes a choice when individual preferences may differ. Clearly, people cannot all have their “ideal” preferences, i.e, the options that they most desire, if those ideal preferences are different. However, for the sake of agreement, people may be willing to accept as a group choice an option that is merely “close” to their ideal preferences. A good example of such a situation is voting for candidates along a political spectrum. We normally think of this spectrum as one-dimensional, with conservative positions on the right and liberal positions on the left, as in Figure 1. While we may represent our ideal preference at some point x on this interval, we might be willing to vote for a candidate that positions himself at some point close to x.