The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).


Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Extreme points of the distance function
on convex surfaces

Author: Tudor Zamfirescu
Journal: Trans. Amer. Math. Soc. 350 (1998), 1395-1406
MSC (1991): Primary 52A15, 53C45
MathSciNet review: 1458314
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus' conjecture that ``almost all" points admit a single farthest point on the surface.

References [Enhancements On Off] (What's this?)

  • 1. A. D. Aleksandrov, Die innere Geometrie der konvexen Flächen, Akademie-Verlag, Berlin, 1955. MR 17:74d
  • 2. H. Busemann, Convex Surfaces, Interscience Publishers, New York, 1958. MR 21:3900
  • 3. Y. Burago, M. Gromov and G. Perelman, A. D. Aleksandrov spaces with curvature bounded below, Russian Math. Surveys 47 (1992), no. 2, 1 - 58. MR 93m:53035
  • 4. J. Cheeger, M. Gromov, C. Okonek and P. Pansu, Geometric Topology: Recent Developments, Lecture Notes in Math. 1504, Springer-Verlag, Berlin, 1991. MR 92m:53001
  • 5. H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved problems in geometry, Springer-Verlag, New York, 1991. MR 92c:52001
  • 6. H. Gluck, D. Singer, Scattering of geodesic fields I, Ann. Math. (2) 108 (1978) 347-372. MR 80c:53046
  • 7. P. Gruber, Baire categories in convexity, in: P. Gruber, J. Wills (eds), Handbook of Convex Geometry, Elsevier Science, Amsterdam, 1993, 1327-1346. MR 94i:52003
  • 8. J. Hebda, Parallel translation of curvature along geodesics, Trans. Amer. Math. Soc. 299 (1987) 559-572. MR 88d:53035
  • 9. J. Hebda, Metric structure of cut loci in surfaces and Ambrose's problem, J. Differential Geometry 40 (1994) 621-642. MR 95m:53046
  • 10. J.-I. Itoh, The length of a cut locus on a surface and Ambrose's problem, J. Differential Geometry 43 (1996) 642-651. MR 97i:53038
  • 11. S. Kobayashi, On conjugate and cut loci, Global Differential Geometry 27 (1989) 140-169.
  • 12. J. Kunze, Der Schnittort auf konvexen Verheftungsflächen, Berlin, Deutscher Verlag der Wissenschaften 1969. MR 42:6760
  • 13. S. B. Myers, Connections between differential geometry and topology I: Simply connected surfaces, Duke Math. J. 1 (1935) 376-391.
  • 14. S. B. Myers, Connections between differential geometry and topology II: Closed surfaces, Duke Math. J. 2 (1936) 95-102.
  • 15. Y. Otsu, T. Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geometry 39 (1994) 629-658. MR 95e:53062
  • 16. S. Saks, Theory of the Integral (2nd Ed.), Dover Publications, 1964. MR 29:4850
  • 17. K. Shiohama, M. Tanaka, Cut loci and distance spheres on Alexandrov surfaces, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), Sém. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 531-559. CMP 97:06
  • 18. C. Vîlcu, private communication.
  • 19. L. Zají\v{c}ek, Porosity and $\sigma$-porosity, Real Analysis Exch. 13 (1987-88) 314-350; 14 (1988-89), 5. MR 89e:26009; MR 89m:26008
  • 20. T. Zamfirescu, Many endpoints and few interior points of geodesics, Inventiones Math. 69 (1982) 253-257. MR 84h:53088
  • 21. T. Zamfirescu, Porosity in convexity, Real Analysis Exch. 15 (1989-90) 424-436. MR 91e:52002
  • 22. T. Zamfirescu, Conjugate points on convex surfaces, Mathematika 38 (1991) 312-317. MR 93e:52005
  • 23. T. Zamfirescu, Baire categories in convexity, Atti Sem. Mat. Fis. Univ. Modena 39 (1991) 139-164. MR 92c:52002
  • 24. T. Zamfirescu, On some questions about convex surfaces, Math. Nachrichten 172 (1995) 313-324. MR 96e:52004
  • 25. T. Zamfirescu, Points joined by three shortest paths on convex surfaces, Proc. Amer. Math. Soc. 123 (1996) 3513-3518. MR 96a:52001
  • 26. T. Zamfirescu, Farthest points on convex surfaces, Math. Zeitschrift, to appear.
  • 27. T. Zamfirescu, Closed geodesic arcs in Aleksandrov spaces, Rend. Circolo Mat. Palermo Suppl. 50 (1997), 425-430.
  • 28. C. Zong, Strange Phenomena in Convex and Discrete Geometry, Springer-Verlag, New York, 1996. CMP 97:03

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 52A15, 53C45

Retrieve articles in all journals with MSC (1991): 52A15, 53C45

Additional Information

Tudor Zamfirescu
Affiliation: Fachbereich Mathematik, Universität Dortmund, 44221 Dortmund, Germany

Received by editor(s): April 17, 1996
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society