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St. Petersburg Mathematical Journal

This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

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

The 2020 MCQ for St. Petersburg Mathematical Journal is 0.68.

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On Tabachnikov’s conjecture
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by A. I. Nazarov and F. V. Petrov
Translated by: A. I. Nazarov
St. Petersburg Math. J. 19 (2008), 125-135
DOI: https://doi.org/10.1090/S1061-0022-07-00989-2
Published electronically: December 17, 2007
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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
  • S. Tabachnikov. The tale of a geometric inequality, MASS colloquium lecture, 2001.
  • Jeffrey C. Lagarias and Thomas J. Richardson, Convexity and the average curvature of plane curves, Geom. Dedicata 67 (1997), no. 1, 1–30. MR 1468858, DOI 10.1023/A:1004912521664
  • A. D. Aleksandrov, Vnutrennyaya Geometriya Vypuklyh Poverhnosteĭ, OGIZ, Moscow-Leningrad, 1948 (Russian). MR 0029518
  • http://mathworld.wolfram.com/SphericalTrigonometry.html
  • http://mathworld.wolfram.com/Triangle.html
  • http://mathworld.wolfram.com/SphericalExcess.html
  • V. A. Zalgaller, On curves with curvature of bounded variation on a convex surface, Mat. Sbornik N.S. 26(68) (1950), 205–214 (Russian). MR 0035043
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Bibliographic Information
  • A. I. Nazarov
  • Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Pr. 28, Staryĭ Peterhof, St. Petersburg 198504, Russia
  • MR Author ID: 228194
  • 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
  • MR Author ID: 689029
  • ORCID: 0000-0003-1693-2745
  • Email: fedorpetrov@mail.ru
  • 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
  • © Copyright 2007 American Mathematical Society
  • 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
  • MathSciNet review: 2319514