Fair division and generalizations of Sperner- and KKM-type results

TL;DR

This paper explores fair division problems, extending classical combinatorial lemmas like Sperner's and KKM to new settings, including polytopes and pseudomanifolds, and establishes related existence results for fair division scenarios.

Contribution

It introduces novel extensions of Sperner's lemma and the KKM theorem to polytopes and pseudomanifolds, improving previous results and connecting fair division with topological combinatorics.

Findings

Extended Alon's necklace splitting result in new regimes

Established fair cake division and rental harmony without full information

Provided quantitative versions of Sperner's lemma and KKM theorem for polytopes

Abstract

We treat problems of fair division, their various interconnections, and their relations to Sperner's lemma and the KKM theorem as well as their variants. We prove extensions of Alon's necklace splitting result in certain regimes and relate it to hyperplane mass partitions. We show the existence of fair cake division and rental harmony in the sense of Su even in the absence of full information. Furthermore, we extend Sperner's lemma and the KKM theorem to (colorful) quantitative versions for polytopes and pseudomanifolds. For simplicial polytopes our results turn out to be improvements over the earlier work of De Loera, Peterson, and Su on a polytopal version of Sperner's lemma. Moreover, our results extend the work of Musin on quantitative Sperner-type results for PL manifolds.