Topological methods in combinatorial geometry

Карасев, Роман

doi:10.1070/rm2008v063n06abeh004578

Cited by 11 publications

(6 citation statements)

References 97 publications

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“…Therefore, is set ∖ is non-empty, the same is true for the right hand side in (29). Thus, Statement (D) of the theorem implies the non-emptiness of sets (28) for any indices (27), i.e., (CSD) is proven. We prove the implication (CSD)⇒(D) under both conditions (F) and (dd).…”

Section: Set-theoretical Differencesmentioning

confidence: 82%

Helly's theorem and shifts of sets. I

Хабибуллин¹

2014

Ufimskii Matematicheskii Zhurnal

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The motivation for the considered geometric problems is the study of conditions under which an exponential system is incomplete in spaces of the functions holomorphic in a compact set and continuous on this compact set. The exponents of this exponential system are zeroes for a sum (finite or infinite) of families of entire functions of exponential type. As is a convex compact set, this problem happens to be closely connected to Helly's theorem on the intersection of convex sets in the following treatment. Let and be two sets in a finite-dimensional Euclidean space being respectively intersections and unions of some subsets. We give criteria for some parallel translation (shift) of set to cover (respectively, to contain or to intersect) set. These and similar criteria are formulated in terms of geometric, algebraic, and set-theoretic differences of subsets generating and .

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Section: Set-theoretical Differencesmentioning

confidence: 82%

Helly's theorem and shifts of sets. I

Хабибуллин¹

2014

Ufimskii Matematicheskii Zhurnal

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“…While Conjecture 13 is still open, here have been many attempts made to solve it. The interested reader can find a number of surveys of the history of these attempts, for example [17,23,27]. This Appendix gives an overview of this history.…”

Section: Appendix a A Brief History Of The Square-peg Problemmentioning

confidence: 99%

“…If we think of the Jordan curve as the "round hole", this conjecture has affectionately been nick-named the square-peg problem by mathematicians. There have been many attempts to resolve the square-peg problem, and a brief overview of the history of the problem can be found in Appendix A, as well as a number of survey articles (see for instance [17,23,27]).…”

Section: Introductionmentioning

confidence: 99%

Square-like quadrilaterals inscribed in embedded space curves

Cantarella,

Denne,

McCleary

2021

Preprint

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The square-peg problem asks if every Jordan curve in the plane has four points which are the vertices of a square. The problem is open for continuous Jordan curves, but it has been resolved for various regularity classes of curves between continuous and C 1 -smooth Jordan curves.Here, in a generalization of the square-peg problem, we consider embedded curves in space, and ask if they have inscribed quadrilaterals with equal sides and equal diagonals. We call these quadrilaterals "square-like". We give a regularity class (finite total curvature without cusps) in which we can prove that every embedded curve has an inscribed square-like quadrilateral. The key idea is to use local data to show that short enough arcs have small curvature, thus ruling out small squares. This allows us to successfully use a limiting argument on approximating curves.Conjecture 1. There is a square-like quadrilateral inscribed in any embedding of S 1 in R n .

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“…His paper opened up the application of topological methods in combinatorics that are now common tools. These techniques appear in several books [132,257] and surveys [62,224]. In many cases the topological methods hinge on the theorems of Brouwer or Borsuk-Ulam; as we discuss in the application sections, on several occasions the topological machinery can be made implicit, and the combinatorial question settled directly by the lemmas of Sperner or Tucker.…”

Section: Combinatorial Topologymentioning

confidence: 99%