A Survey on the Square Peg Problem

Matschke, Benjamin

doi:10.1090/noti1100

Cited by 51 publications

(52 citation statements)

References 49 publications

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“…As a variant of this problem, Meyerson [10] and Kronheimer and Kronheimer [7] proved that for any triangle T and any Jordan curve C, we can find three points on C forming the vertices of a triangle similar to T (note the contrast to our Theorem 2 where we needed the triangle to be equilateral). See a recent survey of Matschke [9] on these problems.…”

Section: Tripodal Pointsmentioning

confidence: 99%

Weight Balancing on Boundaries and Skeletons

Barba

Cheong

Carufel

et al. 2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, i.e., whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any bounded closed region with connected boundary) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional bounded polyhedron containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) On the 1-skeleton of any 3-dimensional bounded convex polyhedron containing the origin, there exist three points whose center of mass coincides with the origin.

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Section: Tripodal Pointsmentioning

confidence: 99%

Weight Balancing on Boundaries and Skeletons

Barba

Cheong

Carufel

et al. 2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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“…There were also high-dimension extensions of these results [4,5,9,6]. For more details we refer the reader to the survey [11] by B. Matschke.…”

Section: Introductionmentioning

confidence: 78%

“…A proof was claimed by H. Griffiths [3], but an error was found later. For a discussion see [11,Conjecture 8]. The specific case of aspect ratio √ 3 and "close to convex" curves was solved B. Matschke [10].…”

Section: Introductionmentioning

confidence: 99%

Any Cyclic Quadrilateral Can Be Inscribed in Any Closed Convex Smooth Curve

Akopyan

Avvakumov

2018

Forum of Mathematics, Sigma

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“…This problem is loosely analogous to the Toeplitz, or square peg, conjecture that every Jordan curve has an inscribed square. This conjecture is proved in particular cases (polygons, smooth curves), but it remains open in the full generality; see [21] for a recent survey. However, a stronger result holds for an arbitrary smooth convex curve γ: there exists a homothetic copy of an arbitrary cyclic quadrilateral inscribed in γ, see [1].…”

Section: Problem 11: Setting a Table On An Uneven Floormentioning

confidence: 99%

Fun Problems in Geometry and Beyond

Khesin

Tabachnikov²

2019

SIGMA

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We discuss fun problems, vaguely related to notions and theorems of a course in differential geometry. This paper can be regarded as a weekend "treasure chest" supplementing the course weekday lecture notes. The problems and solutions are not original, while their relation to the course might be so.Abstract. We discuss fun problems, vaguely related to notions and theorems of a course in differential geometry. This paper can be regarded as a weekend "treasure chest" supplementing the course weekday lecture notes. The problems and solutions are not original, while their relation to the course might be so.

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A Survey on the Square Peg Problem

Cited by 51 publications

References 49 publications

Weight Balancing on Boundaries and Skeletons

Weight Balancing on Boundaries and Skeletons

Any Cyclic Quadrilateral Can Be Inscribed in Any Closed Convex Smooth Curve

Fun Problems in Geometry and Beyond

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