A voting method's win region for a candidate X is the set of elections where X wins.
Most voting methods' win regions can be characterized as unions of convex polytopes. If every candidate's win region is a convex polytope in itself, then the method passes the consistency criterion.
It is also possible to analyze the geometry of win regions to construct possibility or impossibility proofs for broad classes of election methods. Alex Small did so to determine what classes of ranked voting method pass the strong favorite betrayal criterion.
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- Small, Alex (2010-08-25). "Geometric construction of voting methods that protect voters' first choices". arXiv:1008.4331 [cs, math].