Proceedings of the Japan Academy, Series A, Mathematical Sciences

A characterization of convex cones

Kyeonghee Jo

Full-text: Open access

Abstract

Any convex cone has an accumulation point in the base by the action of its automorphism group. In this paper, we prove the converse of this statement, more precisely, a convex domain $\Omega$ with a face $F$ of codimension 1 is a cone over $F$ if there is an Aut($\Omega$)-orbit accumulating at a point of $F$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 10 (2008), 175-178.

Dates
First available in Project Euclid: 2 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.pja/1228226749

Digital Object Identifier
doi:10.3792/pjaa.84.175

Mathematical Reviews number (MathSciNet)
MR2483562

Zentralblatt MATH identifier
1163.57024

Subjects
Primary: 57S20: Noncompact Lie groups of transformations 57N16: Geometric structures on manifolds [See also 57M50] 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 32F45: Invariant metrics and pseudodistances

Keywords
Cone convex domain projective transformation

Citation

Jo, Kyeonghee. A characterization of convex cones. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 10, 175--178. doi:10.3792/pjaa.84.175. https://projecteuclid.org/euclid.pja/1228226749


Export citation

References

  • J.-P. Benzécri, Sur les variétés localement affines et localement projectives, Bull. Soc. Math. France 88 (1960), 229–332.
  • K. Jo, Quasi-homogeneous domains and convex affine manifolds, Topology Appl. 134 (2003), no. 2, 123–146.
  • N. H. Kuiper, On convex locally-projective spaces, in Convegno Internazionale di Geometria Differenziale, Italia, 200–213, Ed. Cremonese, Roma, 1953.
  • J. Vey, Sur les automorphismes affines des ouverts convexes saillants, Ann. Scuola Norm. Sup. Pisa (3) 24 (1970), 641–665.
  • E. B. Vingberg, The theory of convex homogeneous cones, Trans. Moscow Math. Soc. 12 (1963), 340–403.
  • È. B. Vinberg and V. G. Kac, Quasi-homogeneous cones, Mat. Zametki 1 (1967), 347–354.