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

Akopyan, Arseniy; Avvakumov, Sergey

doi:10.1017/fms.2018.7

Cited by 16 publications

(16 citation statements)

References 14 publications

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“…This implies that the points γ(0), γ(a), γ(b), and γ(c) describe a parallelogram inscribed into γ, where the vertex γ(0) was prescribed in advance. Equation (1) ensures that the parallelogram is non-degenerate.…”

Section: Inscribing Parallelograms and Rectanglesmentioning

confidence: 99%

“…5.4] and Schwartz' recent trichotomy of inscribed rectangles [28]. For additional very recent progress on special inscribed quadrilaterals see [1,16,23].…”

Section: Introductionmentioning

confidence: 99%

“…• Any rectifiable simple planar loop inscribes a rectangle that cuts the loop into four parts γ (1) ,…”

Section: Introductionmentioning

confidence: 99%

“…, γ (4) in cyclic order such that the total length of γ (1) and γ (3) is equal to the total length of γ (2) and γ (4) ; see Theorem 3.2.…”

Section: Introductionmentioning

confidence: 99%

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Splitting Loops and Necklaces: Variants of the Square Peg Problem

Aslam

Chen

Frick

et al. 2020

Forum of Mathematics, Sigma

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Toeplitz conjectured that any simple planar loop inscribes a square. Here we prove variants of Toeplitz' square peg problem. We prove Hadwiger's 1971 conjecture that any simple loop in 3-space inscribes a parallelogram. We show that any simple planar loop inscribes sufficiently many rectangles that their vertices are dense in the loop (independently due to Schwartz). If the loop is rectifiable, there is a rectangle that cuts the loop into four pieces that can be rearranged to form two loops of equal length. A rectifiable loop in d-space can be cut into (r − 1)(d + 1) + 1 pieces that can be rearranged by translations to form r loops of equal length.We relate our results to fair divisions of necklaces in the sense of Alon and to Tverberg-type results. This provides a new approach and a common framework to obtain variants of Toeplitz' square peg problem for the class of all continuous curves.

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Section: Inscribing Parallelograms and Rectanglesmentioning

confidence: 99%

“…5.4] and Schwartz' recent trichotomy of inscribed rectangles [28]. For additional very recent progress on special inscribed quadrilaterals see [1,16,23].…”

Section: Introductionmentioning

confidence: 99%

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Splitting Loops and Necklaces: Variants of the Square Peg Problem

Aslam

Chen

Frick

et al. 2020

Forum of Mathematics, Sigma

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“…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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Pushing a Rectangle down a Path

Schwartz

2018

Math Intelligencer

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

Abstract: We prove that any cyclic quadrilateral can be inscribed in any closed convex C 1 -curve. The smoothness condition is not required if the quadrilateral is a rectangle.

Cited by 16 publications

References 14 publications

Splitting Loops and Necklaces: Variants of the Square Peg Problem

Splitting Loops and Necklaces: Variants of the Square Peg Problem

Fun Problems in Geometry and Beyond

Pushing a Rectangle down a Path

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