On some extremal problems in discrete geometry and applications of extremal graphs

Authors

  • Gonchigdorzh Radnaasumberel Mongolian National University, Ulaanbaatar 16060, Mongolia

Keywords:

Polygon, discrete geometry, extremal graph, Turan’s graph, clique, partite graph

Abstract

Introducing the notions of distant pairs of vertices and big subtriangles of a polygon, and using extremal graph theory—specifically, Tutan's graph—we establish upper bounds for:

  • the sum of distances between all distant pairs of vertices in polygons with unit perimeter, and
  • the sum of areas of all big subtriangles in convex polygons with unit area.

We also formulate a conjecture on the upper bound of the sum of the areas of all subtriangles in a convex polygon.

Author correction updated dated on 2025-10-21

Abstract
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Author Biography

Gonchigdorzh Radnaasumberel, Mongolian National University, Ulaanbaatar 16060, Mongolia

Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Ulaanbaatar 13330, Mongolia

References

[1] IMO, "International Mathematical Olympiad, Problems and Solutions, 1959-2009," From:hengsokha.files.wordpress.com.

[2] C. Audet, P. Hansen, and F. Messine, "Extremal problems for convex polygons," Journal of Global Optimization, vol. 38, 2007, pp. 163-179. https://doi.org/10.1007/s10898-006-9065-5

[3] G. Larcher and F. Pillichshammer, "The sum of Distances between Vertices of a Convex

Polygon with unit Perimeter,"The American Mathematical Monthly, 115(4), 2008, pp. 350-355. https://doi.org/10.1080/00029890.2008.11920535

[4] U. Ninjbat, B. Gombodor, and P. Damba, "In the Land of Convex Polygons,"Resonance, vol. 24, 2019, pp. 583-595. https://doi.org/10.1007/s12045-019-0812-6

[5] P. Turan, "On an extremal problem in graph theory," Matematikai es Fizikai Lapok (in Hungarian), vol. 48, 1941, pp. 436-452.

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Published

2024-12-27

How to Cite

On some extremal problems in discrete geometry and applications of extremal graphs. (2024). Journal of Institute of Mathematics and Digital Technology, 6(1), 14-23. https://doi.org/10.5564/jimdt.v6i1.3761

Issue

Vol. 6 No. 1 (2024)

Section

Articles

License

Copyright (c) 2024 Gonchigdorzh Radnaasumberel

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How to Cite

On some extremal problems in discrete geometry and applications of extremal graphs. (2024). Journal of Institute of Mathematics and Digital Technology, 6(1), 14-23. https://doi.org/10.5564/jimdt.v6i1.3761