Abstract: Gerrymandering – the manipulation of electoral boundaries to maximize constituency wins – is often seen as a pathology of democratic systems. A commonly cited cure is to require that electoral constituencies have a “compact” shape. But how much of a constraint does compactness in fact place on would-be gerrymanderers? We operationalize compactness as a convexity constraint and apply a theorem of Kaneko, Kano, and Suzuki (2004) to the two party situation to show that for any population distribution a gerrymanderer can always create equal (population) sized convex constituencies that translate a margin of k voters into a margin of at least k constituency wins. Thus even with a small margin a majority party can win all constituencies. In addition we show that there always exists some population distribution such that all divisions into equal sized convex constituencies translate a margin of k voters into a margin of exactly k constituencies. Thus a convexity constraint can sometimes prevent a gerrymanderer from generating any wins for a minority party. These results clarify that the heart of the problem with outcomes that deviate from proportionality in single member constituency systems is not the manner in which constituencies are drawn but the poverty of the information that is aggregated.