Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

On Tabachnikov's conjecture


Authors: A. I. Nazarov and F. V. Petrov
Translated by: A. I. Nazarov
Original publication: Algebra i Analiz, tom 19 (2007), nomer 1.
Journal: St. Petersburg Math. J. 19 (2008), 125-135
MSC (2000): Primary 53A04; Secondary 52A40, 52A10
DOI: https://doi.org/10.1090/S1061-0022-07-00989-2
Published electronically: December 17, 2007
MathSciNet review: 2319514
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Tabachnikov's conjecture is proved: for any closed curve $ \Gamma$ lying inside a convex closed curve $ \Gamma_1$ the mean absolute curvature $ T(\Gamma)$ exceeds $ T(\Gamma_1)$ if $ \Gamma\ne k\Gamma_1$.


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

  • 1. S. Tabachnikov. The tale of a geometric inequality, MASS colloquium lecture, 2001.
  • 2. J. Lagarias, T. Richardson. Convexity and the average curvature of the plane curves, Geom. Dedicata, 67 (1997), 1-38. MR 1468858 (98f:52007)
  • 3. A.D. Aleksandrov. Intrinsic Geometry of Convex Surfaces. OGIZ, Moscow-Leningrad, 1948; English transl., A.D. Alexandrov. Selected works: Intrinsic Geometry of Convex Surfaces. CRC, 2005. MR 0029518 (10:619c)
  • 4. http://mathworld.wolfram.com/SphericalTrigonometry.html
  • 5. http://mathworld.wolfram.com/Triangle.html
  • 6. http://mathworld.wolfram.com/SphericalExcess.html
  • 7. V. A. Zalgaller. On curves with curvature of bounded variation on a convex surface, Mat. Sbornik, 26(68) (1950), 205-214. (Russian) MR 0035043 (11:681a)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 53A04, 52A40, 52A10

Retrieve articles in all journals with MSC (2000): 53A04, 52A40, 52A10


Additional Information

A. I. Nazarov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Pr. 28, Staryĭ Peterhof, St. Petersburg 198504, Russia
Email: an@AN4751.spb.edu

F. V. Petrov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: fedorpetrov@mail.ru

DOI: https://doi.org/10.1090/S1061-0022-07-00989-2
Keywords: Geometric inequalities, mean absolute curvature, convexity
Received by editor(s): August 1, 2006
Published electronically: December 17, 2007
Additional Notes: Supported by grant NSh-8336.2006.1 (the first author) and by grants NSh-2251.2003.1 and RFFR 05-01-00899 (the second author)
Dedicated: To V. A. Zalgaller on the occasion of his 85th birthday with love and great respect
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society