Cutting Cakes with Combinatorial Fixed Point Theorems

Sperner's Lemma, a combinatorial analogue to Brower's Fixed Point Theorem, guarantees a "fully-labeled cell" in any labeled triangulation of the n-simplex meeting suitable conditions. These fully-labeled cells provide crisp solutions to envy-free cake-cutting questions.  We look at equivalences between combinatorial results and topological fixed-point theorems, along with applications to fair division problems, including the division of multiple cakes and consensus halving. 

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