# Fair Division and Generalizations of Sperner- and KKM-type Results

@article{Asada2017FairDA, title={Fair Division and Generalizations of Sperner- and KKM-type Results}, author={Megumi A Asada and Florian Frick and Vivek Pisharody and Maxwell Polevy and David Stoner and Ling Hei Tsang and Zoe Wellner}, journal={SIAM J. Discret. Math.}, year={2017}, volume={32}, pages={591-610} }

We treat problems of fair division, their various interconnections, and their relations to Sperner's lemma and the Knaster--Kuratowski--Mazurkiewicz (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…

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## References

SHOWING 1-10 OF 33 REFERENCES

### A Polytopal Generalization of Sperner's Lemma

- MathematicsJ. Comb. Theory, Ser. A
- 2002

This work provides two proofs of the following conjecture: a non-constructive proof introducing the notion of a pebble set of a polytope, and a constructive proof using a path-following argument.

### Algorithmic construction of sets for k-restrictions

- Mathematics, Computer ScienceTALG
- 2006

This work addresses k-restriction problems, which unify combinatorial problems of the following type, and offers a generic algorithmic method that yields considerably smaller constructions.

### A constructive proof of a permutation-based generalization of Sperner's lemma

- MathematicsMath. Program.
- 1989

Gale's generalized KKM lemma is derived from the main result and a permutation-based generalization of Brouwer's fixed point theorem is also given.

### Extensions of Sperner and Tucker's lemma for manifolds

- MathematicsJ. Comb. Theory, Ser. A
- 2015

### Rental Harmony: Sperner's Lemma in Fair Division

- Mathematics
- 1999

My friend’s dilemma was a practical question that mathematics could answer, both elegantly and constructively. He and his housemates were moving to a house with rooms of various sizes and features,…

### Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry

- Mathematics
- 2007

A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not…

### Discrete Splittings of the Necklace

- MathematicsMath. Oper. Res.
- 2008

Three new results are reported on the famous necklace theorem of Alon, Goldberg, and West: a direct proof for the case of two thieves and three types of beads, and an efficient constructiveProof for the general case with two thieves.

### Sperner's Lemma

- MathematicsFormaliz. Math.
- 2010

Sperner's Lemma In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner's lemma. The lemma states that for a function…

### Sperner labellings: A combinatorial approach

- MathematicsJ. Comb. Theory, Ser. A
- 2006