Triangle areas in line arrangements

Gábor Damásdi, Leonardo Martínez-Sandoval, Dániel T. Nagy, Zoltán Lóránt Nagy

Research output: Contribution to journalArticlepeer-review

Abstract

A widely investigated subject in combinatorial geometry, originated from Erdős, is the following. Given a point set P of cardinality n in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. What is the maximum number of triangles of unit area, maximum area or minimum area, that can be determined by an arrangement of n lines in the plane? We prove that the order of magnitude for the maximum occurrence of unit areas lies between Ω(n2) and O(n9∕4+ε), for every ε>0. This result is strongly connected to additive combinatorial results and Szemerédi–Trotter type incidence theorems. Next we show an almost tight bound for the maximum number of minimum area triangles. Finally, we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques.

Original languageEnglish
Article number112105
JournalDiscrete Mathematics
Volume343
Issue number12
DOIs
StatePublished - 1 Dec 2020
Externally publishedYes

Keywords

  • Algebraic curves
  • Extremal problems
  • Incidence theorems
  • Unit area triangles

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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