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Convexity and the Average Curvature of Plane Curves

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Abstract

The average curvature of a rectifiable closed curve in R2 is its total absolute curvature divided by its length. If a rectifiable closed curve in R2 is contained in the interior of a convex set D then its average curvature is at least as large as the average curvature of the simple closed curve ∂ D which bounds the convex set.

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References

  1. Banchoff, T.: Critical points and curvature for embedded polyhedra, J. Differential Geom. 1 (1967), 245–256.

    Google Scholar 

  2. Banchoff, T.: Critical points and curvature for embedded polyhedral surfaces, Amer. Math. Monthly 77 (1970), 475–485.

    Google Scholar 

  3. Burago, Yu. D. and Zalgaller, V. A.: Geometric Inequalities, Springer-Verlag, Berlin, 1988.

    Google Scholar 

  4. Cheeger, J., M¨ uller, W. and Schrader, R.: On the curvature of piecewise flat spaces, Comm. Math. Phys. 92 (1984), 405–454.

    Google Scholar 

  5. Cheeger, J., M¨ uller, W. and Schrader, R.: Kinematic and tube formulas for piecewise linear spaces, Indiana Univ. Math. J. 35 (1986), 737–754.

    Google Scholar 

  6. Chern, S. S.: Curves and surfaces in Euclidean space, in C. W. Curtis (ed.), Studies in Global Geometry and Analysis, MAA, 1967, pp. 16–56.

  7. F´ ary, I.: Sur certaines in´ egalit´ es g´ eom´ etriques, Acta Sci. Math. (Szege) 12 (1950), 117–124.

    Google Scholar 

  8. Flaherty, F. J.: Curvature measures for piecewise linear manifolds, Bull. Amer. Math. Soc. 79 (1973), 100–102.

    Google Scholar 

  9. Galperin, G. and Topygo, A.: Moscow Mathematical Olympiads, Education, Moscow, 1986 (Russian) (Problem #33).

  10. Milnor, J.: On the total curvature of knots, Ann. Math. 52 (1950), 248–257.

    Google Scholar 

  11. Richardson, T. J.: Total curvature and intersection tomography, Adv. Math. (to appear).

  12. Santal´o, L. A.: Integral Geometry and Geometric Probability, Addison-Wesley, Reading, Massachusetts, 1976.

    Google Scholar 

  13. Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993.

    Google Scholar 

  14. Steiner, J.: ¨ Uber parallele Fl¨ achen, Jber. Preuss. Akad. Wiss. (1840), 114–118. (See: Gesammelte Wer ke, Vol. II, New York, Chelsea, 1971, pp. 171-177.)

  15. Steinhaus, H.: Length, shape and area, Colloq. Math. 3 (1954), 1–13.

    Google Scholar 

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Authors and Affiliations

  1. AT & T Labs–Research, Florham Park, NJ, 07932, U.S.A.

    Jefferey C. Lagarias

  2. Bell Laboratories, Murray Hill, NJ, 07974, U.S.A.

    Thomas J. Richardson

Authors
  1. Jefferey C. Lagarias
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  2. Thomas J. Richardson
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Lagarias, J.C., Richardson, T.J. Convexity and the Average Curvature of Plane Curves. Geometriae Dedicata 67, 1–30 (1997). https://doi.org/10.1023/A:1004912521664

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