Publication: La topología de los problemas de división: reparto libre de envidia y división consensuada
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2016
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Abstract
En este trabajo se estudian dos problemas de división y la topología que se requiere para resolverlos. Los problemas son el reparto libre de envidia y la división consensuada, y los resultados topológicos involucrados son el Teorema del Punto Fijo de Brouwer y el Teorema de Borsuk-Ulam. Estos teoremas se deducen de sus análogos discretos: el Lema de Sperner y el Lema de Tucker. También analizamos las equivalencias formales entre estos cuatro resultados.
Palabras clave: División justa y libre de envidia, división consensuada, lema de Sperner, teorema del punto fijo de Brouwer, lema débil de Ky Fan, lema de Tucker, teorema de
Borsuk-Ulam
In this work we study two division problems, and the topology behind their solutions. The problems are fair division and consensus division, and the topological results involved are the Brouwer Fixed Point Theorem and the Borsuk-Ulam Theorem. These theorems are deduced from their discrete versions: the Sperner Lemma and the Tucker Lemma. Also we discuss the formal equivalences among these four results
In this work we study two division problems, and the topology behind their solutions. The problems are fair division and consensus division, and the topological results involved are the Brouwer Fixed Point Theorem and the Borsuk-Ulam Theorem. These theorems are deduced from their discrete versions: the Sperner Lemma and the Tucker Lemma. Also we discuss the formal equivalences among these four results
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[1] K. Fan: A generalization of Tucker combinatorial lemma with topological applications.
Ann. of Math. 56 (1952) 431–437.
[2] M. de Longueville: A course in Topological Combinatorics.
Springer, 2013.
[3] Jiri Matousek: Using the Borsuk-Ulam Theorem.
Springer 2003.
[4] T. Prescott, F.E. Su: A constructive proof of Ky Fan’s generalization of Tucker’s lemma. J. Comb. Theory 111 (2005) 2, 257–265