The maximum number of times the same distance can occur among the vertices of a convex \(n\)-gon is \(O(n\log n)\)

DOI10.1006/jcta.2000.3133zbMath0987.52007OpenAlexW1975526544MaRDI QIDQ5937134

Peter Brass, János Pach

Publication date: 30 October 2001

Published in: Journal of Combinatorial Theory. Series A (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1006/jcta.2000.3133

zbMATH Keywords

unit distancesvertices of a convex \(n\)-gon

Mathematics Subject Classification ID

Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) (52B05) Computer graphics; computational geometry (digital and algorithmic aspects) (68U05) Erd?s problems and related topics of discrete geometry (52C10)

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On the number of occurrences of the \(k\)th smallest distance between points in convex position ⋮ Small distances in convex polygons ⋮ Turán problems for edge-ordered graphs ⋮ On the chromatic number of subsets of the Euclidean plane ⋮ On unit distances in a convex polygon ⋮ On locally Gabriel geometric graphs

Cites Work

• A lower bound on the number of unit distances between the vertices of a convex polygon
• The maximum number of unit distances in a convex \(n\)-gon