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Convex Hull of Extreme Points in Flat Riemannian Manifolds

Year 2022, Volume: 15 Issue: 2, 178 - 182, 31.10.2022
https://doi.org/10.36890/iejg.1046707

Abstract

We show that convex hull of extreme points of a closed convex subset of a compact
flat Riemannian manifold is equal to the subset itself.

References

  • [1] Ballmann, W.: Lectures on Spaces of Nonpositive curvature. Brikhauser, Boston, Basel, Berlin, Stuttgart (1985).
  • [2] Bangert, V.: Totally convex sets in complete Riemannian manifolds. J. Differential Geometry. 16, 333-345 (1981). https://doi.org/10.4310/jdg/1214436108
  • [3] Beltagy, M. Shenawy, S.: On the boundary of closed convex sets in En. arxiv:1301.0688v1 [math.MG] 4 Jan (2013).
  • [4] Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume II: Die Gruppen mit einem endlichen Fundamentalbereich. Mathematische Annalen. 72 400-412 (1912). https://doi.org/10.1007/BF01456724
  • [5] Bredon, B.: Introduction to compact transformation groups. Acad Press. New york, London (1972).
  • [6] do Carmo, M. P.: Riemannian Geometry. Brikhauser, Boston, Basel, Berlin (1992).
  • [7] Munkres, J. R.: Topology; a First course. Prentic-Hall (1974).
  • [8] Lay, S. R.: Convex sets and their applications. John Wiley and Sons. Dekker, New York (1982).
  • [9] Shenawy, S.: Convex and Starshaped Sets in Manifolds without Conjugate Points. International Electronic Journal Of Geometry. Volume 12, no. 2, 223-228 (2019). https://doi.org/10.36890/iejg.628087
Year 2022, Volume: 15 Issue: 2, 178 - 182, 31.10.2022
https://doi.org/10.36890/iejg.1046707

Abstract

References

  • [1] Ballmann, W.: Lectures on Spaces of Nonpositive curvature. Brikhauser, Boston, Basel, Berlin, Stuttgart (1985).
  • [2] Bangert, V.: Totally convex sets in complete Riemannian manifolds. J. Differential Geometry. 16, 333-345 (1981). https://doi.org/10.4310/jdg/1214436108
  • [3] Beltagy, M. Shenawy, S.: On the boundary of closed convex sets in En. arxiv:1301.0688v1 [math.MG] 4 Jan (2013).
  • [4] Bieberbach, L.: Über die Bewegungsgruppen der Euklidischen Räume II: Die Gruppen mit einem endlichen Fundamentalbereich. Mathematische Annalen. 72 400-412 (1912). https://doi.org/10.1007/BF01456724
  • [5] Bredon, B.: Introduction to compact transformation groups. Acad Press. New york, London (1972).
  • [6] do Carmo, M. P.: Riemannian Geometry. Brikhauser, Boston, Basel, Berlin (1992).
  • [7] Munkres, J. R.: Topology; a First course. Prentic-Hall (1974).
  • [8] Lay, S. R.: Convex sets and their applications. John Wiley and Sons. Dekker, New York (1982).
  • [9] Shenawy, S.: Convex and Starshaped Sets in Manifolds without Conjugate Points. International Electronic Journal Of Geometry. Volume 12, no. 2, 223-228 (2019). https://doi.org/10.36890/iejg.628087
There are 9 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Reza Mirzaie 0000-0002-9480-0222

Omid Rezaie This is me 0000-0002-9101-1092

Early Pub Date July 23, 2022
Publication Date October 31, 2022
Acceptance Date May 30, 2022
Published in Issue Year 2022 Volume: 15 Issue: 2

Cite

APA Mirzaie, R., & Rezaie, O. (2022). Convex Hull of Extreme Points in Flat Riemannian Manifolds. International Electronic Journal of Geometry, 15(2), 178-182. https://doi.org/10.36890/iejg.1046707
AMA Mirzaie R, Rezaie O. Convex Hull of Extreme Points in Flat Riemannian Manifolds. Int. Electron. J. Geom. October 2022;15(2):178-182. doi:10.36890/iejg.1046707
Chicago Mirzaie, Reza, and Omid Rezaie. “Convex Hull of Extreme Points in Flat Riemannian Manifolds”. International Electronic Journal of Geometry 15, no. 2 (October 2022): 178-82. https://doi.org/10.36890/iejg.1046707.
EndNote Mirzaie R, Rezaie O (October 1, 2022) Convex Hull of Extreme Points in Flat Riemannian Manifolds. International Electronic Journal of Geometry 15 2 178–182.
IEEE R. Mirzaie and O. Rezaie, “Convex Hull of Extreme Points in Flat Riemannian Manifolds”, Int. Electron. J. Geom., vol. 15, no. 2, pp. 178–182, 2022, doi: 10.36890/iejg.1046707.
ISNAD Mirzaie, Reza - Rezaie, Omid. “Convex Hull of Extreme Points in Flat Riemannian Manifolds”. International Electronic Journal of Geometry 15/2 (October 2022), 178-182. https://doi.org/10.36890/iejg.1046707.
JAMA Mirzaie R, Rezaie O. Convex Hull of Extreme Points in Flat Riemannian Manifolds. Int. Electron. J. Geom. 2022;15:178–182.
MLA Mirzaie, Reza and Omid Rezaie. “Convex Hull of Extreme Points in Flat Riemannian Manifolds”. International Electronic Journal of Geometry, vol. 15, no. 2, 2022, pp. 178-82, doi:10.36890/iejg.1046707.
Vancouver Mirzaie R, Rezaie O. Convex Hull of Extreme Points in Flat Riemannian Manifolds. Int. Electron. J. Geom. 2022;15(2):178-82.