Rad HAZU, Matematičke znanosti, Vol. 27 (2023), 167-173.

REFINED EULER'S INEQUALITIES IN PLANE GEOMETRIES AND SPACES

Darko Veljan

Department of Mathematics, Faculty of Science, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: darko.veljan@gmail.com

Abstract.   Refined famous Euler's inequalities R ≥ nr of an n-dimensional simplex for n = 2, 3 and 4 as well as of non-Euclidean triangles in terms of symmetric functions of edge lengths of a triangle or a simplex in question are shown. Here R is the circumradius and r the inradius of the simplex. We also provide an application to geometric probabilities of our results and an example from astrophysics to the position of a planet within the space of four stars. We briefly discuss a recursive algorithm to get similar inequalities in higher dimensions.

2020 Mathematics Subject Classification.   51M04, 51M09, 51M16, 60D05.

Key words and phrases.   Triangle and tetrahedron inequalities, Euler's inequality in 2D, 3D and 4D, non-Euclidean Euler's inequality, geometric probability applied to astrophysics, simplex inequalities.

Full text (PDF) (free access)

DOI: https://doi.org/10.21857/y6zolb6ldm

References:

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  5. D. Veljan, Refinements of Euler's inequalities in plane, space and n-space, in: Proc. 3th Croatian Combinatorial Days, Zagreb, September 21-22, 2020 (Eds. T. Došlić and S. Majstorović), pp. 129-140.

  6. D. Veljan, Improved Euler's inequalities in plane and space, J. Geom. 112 (2021), no.3, Paper No.31, 11 pp.
    MathSciNet     CrossRef

  7. D. Veljan, Planets are (very likely) in orbits of stars, in: Proc. 4th Croatian Combinatorial Days, Zagreb, September 22-23, 2022 (Eds. T. Došlić, S. Majstorović and L. Podrug), pp. 147-151.

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