Abstract
A mixed symmetric Chernoff type inequality for two convex domains can be proved, furthermore its stability properties can also be obtained in the Hausdorff and \(L_2\) metrics, respectively.
Similar content being viewed by others
References
Bonnesen, T., Fenchel, W.: Theorie der Convxen Körper. Chelsea Publishing, New York (1948)
Böröczky, K.J.: Stability of the Blaschke-Santaló, and the affine isoperimetric inequalities. Adv. Math. 225, 1914–1928 (2010)
Burago, Y.D., Zalgaller, V.A.: Geometric Inequalities. Springer, Berlin (1988)
Chernoff, P.R.: An area-width inequality for convex curves. Am. Math. Monthly 76, 34–35 (1969)
Cufí, J., Gallego, E., Reventós, A.: A note on Hurwitz’s inequality. J. Math. Anal. Appl. 458, 436–451 (2018)
Edler, F.: Vervollständigung der Steiner’schen elementar-geometrischen Beweise für den Satz, dass der Kreis grösseren Flächeninhalt besitzt als jedeandere ebene Figur gleich grossen Umfanges. G“ott. N., 73–80 (1882) (translated into French and printed in Bull. Sci. Math., 7(2), 198–204 (1883))
Figalli, A., Indrei, E.: A sharp stability result for the relative isoperimetric inequality inside convex cones. J. Geom. Anal. 23, 938–969 (2013)
Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182, 167–211 (2010)
Fuglede, B.: Stability in the isoperimetric problem. Bull. Lond. Math. Soc. 18, 599–605 (1986)
Fusco, N.: The classical isoperimetric theorem, https://www.docenti.unina.it/webdocenti-be/allegati/materiale-didattico/377226
Fusco, N., Maggi, F., Pratelli, A.: The sharp quantitative isoperimetric inequality. Ann. Math. 168, 941–980 (2008)
Gao, L., Wang, Y.L.: Stability properties of the generalized Chernoff inequality. Math. Inequal. Appl. 15, 281–287 (2012)
Gardner, R.J.: Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39, 355–405 (2002)
Groemer, H.: Stability theorems for convex domains of constant width. Can. Math. Bull. 31, 328–337 (1988)
Groemer, H.: Stability properties of geometric inequalities. Am. Math. Monthly 97, 382–394 (1990)
Groemer, H.: Stability theorems for two measures of symmetry. Discrete Comput. Geom. 24, 301–311 (2000)
Groemer, H., Schneider, R.: Stability estimates for some geometric inequalities. Bull. Lond. Math. Soc. 23, 67–74 (1991)
Hurwitz, A.: Sur quelques applications géométriques des séries de Fourier. Ann. Sci. I’É. N.S., 19(3e série), 357–408 (1902)
Indrei, E.: A sharp lower bound on the polygonal isoperimetric deficit. Proc. Am. Math. Soc. 144, 3115–3122 (2016)
Indrei, E., Nurbekyan, L.: On the stability of the polygonal isoperimetric inequality. Adv. Math. 276, 62–86 (2015)
Lutwak, E.: Mixed width-integrals of convex bodies. Isr. J. Math. 28, 249–253 (1977)
Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182–1238 (1978)
Osserman, R.: Bonnesen-style isoperimetric inequalities. Am. Math. Monthly 86(1), 1–29 (1979)
Ou, K., Pan, S.L.: Some remarks about closed convex curves. Pacif. J. Math. 2, 393–401 (2010)
Pan, S.L., Xu, H.P.: Stability of a reverse isoperimetric inequality. J. Math. Anal. Appl. 350, 348–353 (2009)
Pan, S.L., Zhang, H.: A reverse isoperimetric inequality for convex plane curves. Beiträge Algebra Geom. 48, 303–308 (2007)
Schneider, R.: A stability estimate for the Aleksandrov–Fenchel inequality, with an application to mean curvature. Manuscr. Math. 69, 291–300 (1990)
Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory(second, expanded edn. Cambridge University Press, Canbridge (2013)
Steiner, J.: Sur le maximum et le minimum des figures dans le plan, sur la sphère, et dans l’espace en général, I and II. J. Reine Angew. Math. (Crelle), 24, 93–152 and 189–250 (1842)
Wang, P., Xu, W., Zhou, J., Zhu, B.: On Bonnesen-style symmetric mixed inequality of two planar convex domains. Sci. China Math. 45, 245–254 (2015)
Xu, W., Zhou, J., Zhu, B.: Bonnesen-style symmetric mixed isoperimetric inequality. In: Real and Complex Submanifold, Springer Proceedings in Mathematics and Statistics, vol. 106, pp. 97–107 (2014)
Zeng, C., Zhou, J., Yue, S.: The symmetric mixed isoperimetric inequality of two planar convex domains. Acta Math. Sin. 55, 355–362 (2012)
Zhang, D.Y.: Inequalities with curvature and their stability estimates for convex curves. J. Math. Inequal. 10, 433–443 (2016)
Zwierzyński, M.: The improved isoperimetric inequality and the Wigner caustic of planar ovals. J. Math. Anal. Appl. 442, 726–739 (2016)
Acknowledgements
The author would like to thank the referees for their suggestions. The author would also like to thank Professor Shengliang Pan for useful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by the National Natural Science Foundation of China (No. 11671298), Natural Science Foundation of Anhui Province (No. 1908085MA05) and University Natural Science Research Project of Anhui Province (No. KJ2019A0590).
Rights and permissions
About this article
Cite this article
Zhang, D. A Mixed Symmetric Chernoff Type Inequality and Its Stability Properties. J Geom Anal 31, 5418–5436 (2021). https://doi.org/10.1007/s12220-020-00485-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-020-00485-0