Convex Hull of Extreme Points in Flat Riemannian Manifolds

Year 2022, Volume: 15 Issue: 2, 178 - 182, 31.10.2022

Reza Mirzaie Omid Rezaie

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.

Keywords

Flat Reimannian manifold, convex subset, convex hull, extreme point

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