St. Petersburg Mathematical Society

Author(s): A. I. Nazarov; 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
Posted: 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 $\Gamma\ne k\Gamma_1$ .

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)

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

A. I. Nazarov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii Pr. 28, Staryi 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: 10.1090/S1061-0022-07-00989-2
PII: S 1061-0022(07)00989-2
Keywords: Geometric inequalities, mean absolute curvature, convexity
Received by editor(s): 1/AUG/2006
Posted: 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 of article: Copyright 2007, American Mathematical Society

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ISSN: 1547-7371(e) ISSN: 1061-0022(p)

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