Abstract
In this paper the classical Banchoff–Pohl inequality, an isoperimetric inequality for nonsimple closed curves in the Euclidean plane, involving the square of the winding number, is generalized to symmetric Minkowski geometries. The proof uses the well-known curve shortening flow.
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References
Angenent, S.: Parabolic equations for curves on surfaces. Part I. Curves with p-integrable curvature, Ann. Math. 132 (1990) 451–483.
Angenent, S.: Parabolic equations for curves on surfaces. Part II. Intersections, blow-up and generalized solutions, Ann. Math. 133 (1991) 171–215.
Banchoff, T. F. and Pohl, W. F.: A generalization of the isoperimetric inequality, J. Diff. Geom. 6 (1971), 171–192.
Benson, R. V.: Euclidean Geometry and Convexity, New York: McGraw-Hill, 1966.
Busemann, H.: The isoperimetric problem in the Minkowski plane, Am. J. Math. 69 (1947), 863–871.
Busemann, H.: The foundations of Minkowskian Geometry. Comment, Math. Helv. 24 (1950), 156–187.
Chou, K. and Zhu, X.: The Curve Shortening Problem, Boca Raton: Chapman & Hall/CRC, 2001.
Gage, M. E.: An isoperimetric inequality with applications to curve shortening, Duke Math. J. 50 (1983), 1225–1229.
Gage, M. E.: Evolving plane curves by curvature in relative geometries, Duke Math. J. 72 (1993), 441–466.
Gage, M. E.: Minkowski Plane Geometry and Anisotropic Curvature Flows of Curves. Curvature flows and related topics (Levico, 1994), 83–87, GAKUTO Intern. Ser. Math. Sci. Appl. 5, Tokyo, 1995.
Gage, M. E. and Hamilton, R. S.: The heat equation shrinking convex plane curves, J. Diff. Geom. 23 (1986), 69–96.
Gage, M. E. and Li, Y.: Evolving plane curves by curvature in relative geometries. II, Duke Math. J. 75 (1994), 79–98.
Grayson, M. A.: The heat equation shrinks embedded plane curves to round curves, J. Diff. Geom. 26 (1987), 285–314.
Guggenheimer, H.: Pseudo-Minkowski differential geometry, Ann. Pura Appl. IV. Ser. 70 (1965), 305–370.
Leichtweiss, K.: Konvexe Mengen, Berlin, Heidelberg, New York: Springer, 1980.
Santaló, L. A.: Integral Geometry and Geometric Probability, Reading, MA: Addison-Wesley, 1976.
Süssmann, B.: Curve shortening flow and the Banchoff–Pohl inequality on surfaces of nonpositive curvature, Contr. Alg. Geom. 40 (1999), 203–215.
Thompson, A. C.: Minkowski Geometry, Encycl. Math. Sci. Appl., Vol. 63, Cambridge University Press, 1996.
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Su˙ssmann, B. Curve Shortening and the Banchoff–Pohl Inequality in Symmetric Minkowski Geometries. Ann Glob Anal Geom 29, 323–332 (2006). https://doi.org/10.1007/s10455-005-9005-5
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DOI: https://doi.org/10.1007/s10455-005-9005-5