Abstract
In this paper, we give a reverse analog of the Bonnesen-style inequality of a convex domain in the surface \( \mathbb{X} \) ∈ of constant curvature ∈, that is, an isoperimetric deficit upper bound of the convex domain in \( \mathbb{X} \) ∈ . The result is an analogue of the known Bottema’s result of 1933 in the Euclidean plane \( \mathbb{E} \) 2.
Similar content being viewed by others
References
Blaschke W. Kreis und Kugel. Berlin: Auflage, 1956
Bottema O. Eine obere Grenze für das isoperimetrische Defizit ebener Kurven. Nederl Akad Wetensch Proc, 1933, 36: 442–446
Burago Y D, Zalgaller V A. Geometric Inequalities. Berlin: Springer-Verlag, 1988
Federer H, Fleming W. Normal and integral currents. Ann of Math, 1960, 72: 458–520
Grinberg E, Ren D, Zhou J. The symmetric isoperimetric deficit and the containment problem in a plane of constant curvature. Preprint
Howard R. Blaschke’s rolling theorem for manifolds with boundry. Manuscripta Math, 1999, 99: 471–483
Klain D. Bonnesen-type inequalities for surfaces of constant curvature. Adv Appl Math, 2007, 39: 143–154
Osserman R. The isoperimetric inequality. Bull Amer Math Soc, 1978, 84: 1182–1238
Osserman R. Bonnesen-style isoperimetric inequality. Amer Math Monthly, 1979, 86: 1–29
Ren D. Topics in Integral Geometry. Singapore: World Scientific, 1994
Santaló L A. Integral Geometry and Geometric Probability. Massachusetts: Addison-Wesley, Adevanced Book Program Reading, 1976
Zhang G. Geometric inequalities and inclusion measures of convex bodies. Mathematika, 1994, 41: 95–116
Zhang G. The affine Sobolev inequality. J Diff Geom, 1999, 53: 183–202
Zhang G, Zhou J. Containment measures in integral geometry. In: Integral Geometry and Convexity. Singapore: World Scientific, 2005
Zhou J. On Bonnesen-type inequalities (in Chinese). Acta Math Sinica Chinese Ser, 2007, 50: 1397–1402
Zhou J. On the Willmore deficit of convex surface. Lectures in Appl Math of Amer Math Soc, 1994, 8: 279–287
Zhou J. Tne Willmore inequalities for submanifolds. Canad Math Bull, 2007, 50: 474–480
Zhou J. The Willmore functional and the containment problem in R 4. Sci China Ser A, 2007, 50: 325–333
Zhou J, Chen F. The Bonnesen-type inequality in a plane of constant cuvature. J Korean Math Soc, 2007, 44: 1363–1372
Zhou J, Zhou C, Ma F. Isoperimetric deficit upper limit of a planar convex set. Rendiconti del Circolo Matemetico di Palermo Serie II, 2009, 81: 363–367
Zhou J, Xia Y, Zeng C. Some new Bonnesen-style inequalities. J Korean Math Soc, to appear
Zhou J, Jiang D, Li M, Chen F. On Ros theorem for hypersurface (in Chinese). Acta Math Sinica Chinese Ser, 2009, 52: 1075–1084
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, M., Zhou, J. An isoperimetric deficit upper bound of the convex domain in a surface of constant curvature. Sci. China Math. 53, 1941–1946 (2010). https://doi.org/10.1007/s11425-010-4018-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-010-4018-3