A Note on Toeplitz’ Conjecture

In 1911, Toeplitz made a conjecture asserting that every Jordan curve in contains four points forming the corners of a square. Here Conjecture C is presented, which states that the side length of the largest square on a closed curve that consists of edges of an n × n grid is at least times the side...

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Bibliographic Details
Published in:	Discrete & computational geometry Vol. 51; no. 3; pp. 722 - 728
Main Authors:	Pettersson, Ville H., Tverberg, Helge A., Östergård, Patric R. J.
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01-04-2014 Springer Nature B.V
Subjects:	Combinatorics Computation Computational geometry Computational Mathematics and Numerical Analysis Corners Forming Geometry Mathematics Mathematics and Statistics Texts Theorems Grid graph Toeplitz’ conjecture Inscribed square Jordan curve Chordless cycle Depth-first search
Online Access:	Get full text
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Summary:	In 1911, Toeplitz made a conjecture asserting that every Jordan curve in contains four points forming the corners of a square. Here Conjecture C is presented, which states that the side length of the largest square on a closed curve that consists of edges of an n × n grid is at least times the side length of the largest axis-aligned square contained inside the curve. Conjecture C implies Toeplitz’ conjecture and is verified computationally for n ≤13.
Bibliography:	ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23
ISSN:	0179-5376 1432-0444
DOI:	10.1007/s00454-014-9578-5