Generalized Sperner lemma and subdivisions into simplices of equal volume

Bekker, Boris M.; Netsvetaev, N. Yu.

doi:10.1007/bf02434927

Cited by 4 publications

(6 citation statements)

References 7 publications

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“…In 1994 Bekker and Netsvetaev proved similar statement in higher dimensions [1]. 1 By the phrase polygon B is cut into triangles we mean that B can be presented as a union of a finite number of triangles so that the interiors of the triangles have an empty intersection with each other. Fig.…”

Section: Equidissection Problemmentioning

confidence: 93%

On equidissection of balanced polygons

Rudenko

2013

J Math Sci

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In this paper we show that a lattice balanced polygon of odd area cannot be cut into an odd number of triangles of equal areas. First result of this type was obtained by Paul Monsky in 1970. He proved that a square cannot be cut into an odd number of triangles of equal areas. In 2000 Sherman Stein conjectured that the same holds for any balanced polygon.We also show connections between the equidissection problem and tropical geometry. arXiv:1206.4591v2 [math.CO]

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Section: Equidissection Problemmentioning

confidence: 93%

On equidissection of balanced polygons

Rudenko

2013

J Math Sci

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“…Sperner's Lemma for polytopes. We now describe a more general Sperner's Lemma-type of result for polytopes, following [4]. A related result can be found in [16].…”

Section: 2mentioning

confidence: 99%

“…Remark 1. By taking into account the orientation of the polytope, we get here a definition of the index slightly different from the one in [4], which uses Z 2 indices in the above recursive definition. We note that the definition of the index in [4] is inconsistent with some of the results later in that paper, e.g., with Proposition 1 quoted below.…”

Section: 2mentioning

confidence: 99%

Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics

Gidea¹,

Shmalo²

2018

Discrete &Amp; Continuous Dynamical Systems - A

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We present a combinatorial approach to rigorously show the existence of fixed points, periodic orbits, and symbolic dynamics in discrete-time dynamical systems, as well as to find numerical approximations of such objects. Our approach relies on the method of 'correctly aligned windows'. We subdivide 'windows' into cubical complexes, and we assign to the vertices of the cubes labels determined by the dynamics. In this way, we encode the information on the dynamics into combinatorial structure. We use a version of Sperner's Lemma to infer that, if the labeling satisfies certain conditions, then there exist fixed points/periodic orbits/orbits with prescribed itineraries. The method developed here does not require the computation of algebraic topologytype invariants, as only combinatorial information is needed; our arguments are elementary.

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“…In this section, we describe a more general Sperner lemma-type of result for convex polytopes (which shall be referred to as polytopes), following [BN98]. We use the result to prove another generalization of Sperner's lemma, Theorem 3.4, one which can be directly used to prove Kakutani's theorem.…”

Section: Polytope Version Of Sperner's Lemmamentioning

confidence: 99%

“…The following is a generalization of Sperner's lemma from [BN98]. We provide the proof for clarification and to ensure that the upcoming proof for Theorem 3.4 is intuitive.…”

Section: Proposition 31 ([Bn98]mentioning

confidence: 99%

Combinatorial Proof of Kakutani's Fixed Point Theorem

Shmalo¹

2018

Preprint

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Kakutani's fixed point theorem is a generalization of Brouwer's fixed point theorem to upper semicontinuous multivalued maps and is used extensively in game theory and other areas of economics. Earlier works have shown that Sperner's lemma implies Brouwer's theorem. In this paper, a new combinatorial labeling lemma, generalizing Sperner's original lemma, is given and is used to derive a simple proof for Kakutani's fixed point theorem. The proof is constructive and can be easily applied to numerically approximate the location of fixed points. The main method of the proof is also used to obtain a generalization of Kakutani's theorem for discontinuous maps which are locally gross direction preserving.

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Generalized Sperner lemma and subdivisions into simplices of equal volume

Cited by 4 publications

References 7 publications

On equidissection of balanced polygons

On equidissection of balanced polygons

Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics

Combinatorial Proof of Kakutani's Fixed Point Theorem

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