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

  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.},
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… 

Figures from this paper

Multilabeled versions of Sperner's and Fan's lemmas and applications

A general technique related to the polytopal Sperner lemma is proposed for proving old and new multilabeled versions of Sperer's lemma, which yields a cake-cutting theorem where the number of players and the numberof pieces can be independently chosen.


We propose a general technique related to the polytopal Sperner lemma for proving old and new multilabeled versions of Sperner’s lemma. A notable application of this technique yields a cake-cutting

Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals

We prove a common strengthening of Bárány’s colorful Carathéodory theorem and the KKMS theorem. In fact, our main result is a colorful polytopal KKMS theorem, which extends a colorful KKMS theorem

Different versions of the nerve theorem and rainbow simplices.

Given a simplicial complex and a collection of subcomplexes covering it, the nerve theorem, a fundamental tool in topological combinatorics, guarantees a certain connectivity of the simplicial

Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals

A common strengthening of B\'ar\'any's colorful Carath\'eodory theorem and the KKMS theorem is proved and an upper bound on the piercing number of colorful d-interval hypergraphs is established.

Envy-free division via configuration spaces

The classical approach to envy-free division and equilibrium problems arising in mathematical economics typically relies on KnasterKuratowski-Mazurkiewicz theorem, Sperner’s lemma or some extension

Thieves can make sandwiches

We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of m measures in Rd among r thieves using


Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz’s square peg problem. We prove Hadwiger’s 1971 conjecture that any simple loop in $3$-space

Envy-free cake division without assuming the players prefer nonempty pieces

The main step in the proof is a new combinatorial lemma in topology, close to a conjecture by Segal-Halevi and reminiscent of the celebrated Sperner lemma: instead of restricting the labels that can appear on each face of the simplex, the lemma considers labelings that enjoy a certain symmetry on the boundary.



A Polytopal Generalization of Sperner's Lemma

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

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

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

  • O. Musin
  • Mathematics
    J. Comb. Theory, Ser. A
  • 2015

Rental Harmony: Sperner's Lemma in Fair Division

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

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

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

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