Abstract

In Guo and Peng's article [Spherically convex sets and spherically convex functions, J. Convex Analysis 28 (2021) 103–122] the authors define the notions of spherical convex sets and functions on "general curved surfaces" in Rn\mathbb{R}^n (n ≥ 2), they study several properties of these classes of sets and functions, and they establish analogues of Radon, Helly, Carathéodory and Minkowski theorems for spherical convex sets, as well as some properties of spherical convex functions which are analogous to those of usual convex functions. In obtaining such results, the authors use an analytic approach based on their definitions. Our aim in this note is to provide simpler proofs for these results on spherical convex sets; our proofs are based on some characterizations/representations of spherical convex sets by usual convex sets in Rn\mathbb{R}^n. Moreover, we provide a new proof of the recent result of Han and Nashimura on the separation of spherical convex sets established in arXiv:2002.06558. Our proof is based on a result stated in locally convex spaces.

Author information

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

Constantin Zălinescu

Octav Mayer Institute of Mathematics, Romanian Academy of Sciences, Iaşi, Romania
and: University "Alexandru Ioan Cuza", Iaşi, Romania

zalinesc@uaic.ro

Keywords

  • Spherical convexity
  • Helly theorem
  • Carathéodory theorem
  • separation theorem

Mathematics Subject Classification

52A55, 52A20, 52A35

References

  1. I. J. Bakelman: Convex Analysis and Nonlinear Geometric Elliptic Equations, Springer, Berlin (1994).https://doi.org/10.1007/978-3-642-69881-1Google Scholar
  2. M. J. C. Baker: A Helly-type theorem on a sphere, J. Austral. Math. Soc. 7 (1967) 323–326.https://doi.org/10.1017/S144678870000416XGoogle Scholar
  3. L. Danzer, B. Grünbaum, V. Klee: Helly’s theorem and its relatives, in: Proc. Symp. Pure Math., Vol. VII, American Mathematical Society, Providence (1963) 101–180.https://doi.org/10.1090/pspum/007/0157289Google Scholar
  4. O. P. Ferreira, A. N. Iusem, S. Z. Németh: Projections onto convex sets on the sphere, J. Global Optim. 57/3 (2013) 663–676.https://doi.org/10.1007/s10898-012-9914-3Google Scholar
  5. Q. Guo, Y. Peng: Spherically convex sets and spherically convex functions, J. Convex Analysis 28/1 (2021) 103–122.https://doi.org/10.68381/jca28009Google Scholar
  6. H. Han, T. Nishimura: Spherical separation theorem, arXiv: 2002.06558 (2020).Google Scholar
  7. W. Hansen, V. Klee: Intersection theorems for positive sets, Proc. Amer. Math. Soc. 22 (1969) 450–457.https://doi.org/10.1090/S0002-9939-1969-0254563-2Google Scholar
  8. G. Köthe: Topological Vector Spaces I, Springer, New York (1969).Google Scholar
  9. M. Lassak: Width of spherical convex bodies, Aequationes Math. 89/3 (2015) 555–567.https://doi.org/10.1007/s00010-013-0237-3Google Scholar
  10. J. Molnár: Über eine Übertragung des Hellyschen Satzes in sphärische Räume, Acta Math. Acad. Sci. Hungar. 8 (1957) 315–318.https://doi.org/10.1007/BF02020320Google Scholar
  11. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  12. R. Schneider: Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press, Cambridge (1993).https://doi.org/10.1017/CBO9780511526282Google Scholar
  13. L. G. Šaraburova, J. A. Šaškin: Intersections of convex cones, Mat. Zametki 24/3 (1978) 391–402; English translation: Math. Notes 24/3-4 (1979) 708–714.https://doi.org/10.1007/BF01097762Google Scholar

Suggested citation

C. Zălinescu. “New Proofs for some Results on Spherically Convex Sets.” Journal of Convex Analysis 29 (2022), No. 4, 995–1006. https://doi.org/10.68381/jca29056

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Published by Heldermann Verlag, 2022. Rights now held by Banach Press.