Abstract

Let ∂ C\partial \,\mathcal{C} be the boundary of a compact convex body C\mathcal{C} in Rn, n≥2\mathbb{R}^n,\, n\geq 2, and OO be an interior point of C\mathcal C. Every straight line ll containing OO cuts from C\mathcal{C} a segment [AB][AB] with end-points on ∂ C\partial \,\mathcal{C}. It is shown that if [AB][AB] is the shortest such segment, then ∂ C\partial \,\mathcal{C} is smooth at the points AA and BB (i.e. at both of them there is only one supporting hyperplane for C\mathcal{C}) and, something more, the normals to the unique supporting hyperplanes at the points AA and BB intersect at a point belonging to the hyperplane through OO which is orthogonal to [AB][AB].
If C\mathcal{C} is a smooth compact convex body in Rn, n≥2\mathbb{R}^n,\, n\geq 2, the above property holds also when [AB][AB] is the longest such segment. Similar results are also valid when OO is outside the set C\mathcal{C}. The ``local versions'' of these results (when the length ∣AB∣|AB| of the segment [AB][AB] is locally maximal or locally minimal) are valid as well. More specific results are obtained in the particular case when C\mathcal{C} is a convex polytope

Author information

Contact details are reproduced from the original publication and may be historical.

Petar Kenderov

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

vorednek@gmail.com

Oleg Mushkarov

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

muskarov@math.bas.bg

Nikolai Nikolov

(1) Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
(2) Faculty of Information Sciences, State University of Library Studies and Information Technologies, Sofia, Bulgaria

nik@math.bas.bg

Keywords

  • Convex set
  • extremal problem
  • Philo's line
  • critical point
  • convex polytope
  • concurrent lines

Mathematics Subject Classification

51M16, 52A20, 52A40, 49K05, 49K10

References

  1. O. Bottema, Verscheidenheden, Euclides, Maanblad voor de Didaktiek van de Wiskunde (1968) 20–25.Google Scholar
  2. J. Casey: A Sequel to the First Six Books of the Elements of Euclid, Hodges, Figis, and Co., Dublin (1886).Google Scholar
  3. H. S. M. Coxeter: Introduction to Geometry, Wiley, New York (1961).Google Scholar
  4. H. S. M. Coxeter, J. van de Craats: Philon lines in non-Eucledean planes, J. Geometry 48 (1993) 26–55.https://doi.org/10.1007/BF01226799Google Scholar
  5. H. Eves: Philo’s line, Scripta Mathematica 24 (1959) 141–148.https://doi.org/10.1080/00049185908702338Google Scholar
  6. P. S. Kenderov: Powering knowledge versus pouring facts, in: Invited Lectures from the 13th International Congress on Mathematical Education, G. Kaiser et al. (eds.), ICME -13, Springer Open, Berlin (2018) 289–306.https://doi.org/10.1007/978-3-319-72170-5_17Google Scholar
  7. S. L’Huilier: Principiorum calculi differentialis et integralis. Expositio elementaris, J. G. Cotta, Tubingae (1795) 270.Google Scholar
  8. S. L’Huilier: Solution d’un problème de géométrie, dépendant de la théorie des maximis et minimis, Ann. Math. Pures Appl. 2 (1811-1812) 17–22.Google Scholar
  9. E. Neovius: Ueber eine specielle geometrische Aufgabe des Minimums, Nachrichten d. Königl. Gesellschaft d. Wissenschaften zu Göttingen 14 (1887) 359–361.https://doi.org/10.1007/BF01206220Google Scholar
  10. J. Neuberg: Sur un minimum, Mathesis (1907) 68–69.Google Scholar
  11. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  12. Wikipedia: http://en.wikipedia.org/wiki/Philo-of-Byzantium, retrieved on August 21 (2024).Google Scholar
  13. Wikipedia: https://en.wikipedia.org/wiki/Philo line, retrieved on August 21 (2024).Google Scholar

Suggested citation

P. Kenderov, O. Mushkarov, N. Nikolov. “A Generalization of a Classical Geometric Extremum Problem.” Journal of Convex Analysis 34 (2027), No. 1, 33–48. https://doi.org/10.68381/jca34004

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