Bonnesen-style Wulff isoperimetric inequality
- Zengle Zhang^{1} and
- Jiazu Zhou^{1, 2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-017-1305-3
© The Author(s) 2017
Received: 25 November 2016
Accepted: 23 January 2017
Published: 14 February 2017
Abstract
The Wulff isoperimetric inequality is a natural extension of the classical isoperimetric inequality (Green and Osher in Asian J. Math. 3:659-676 1999). In this paper, we establish some Bonnesen-style Wulff isoperimetric inequalities and reverse Bonnesen-style Wulff isoperimetric inequalities. Those inequalities obtained are extensions of known Bonnesen-style inequalities and reverse Bonnesen-style inequalities.
Keywords
MSC
1 Introduction and main results
Many \(B_{K}\)s were found in the last century, and mathematicians are still working on those unknown isoperimetric deficit lower limits of geometric significance. For more details, see references [20, 24–26].
If W is the unit disc, then \(r_{W}\) and \(R_{W}\) are, respectively, the radius of the maximum inscribed disc and the radius of the minimum circumscribed disc of K.
Our main results are the following.
Theorem 1
Theorem 2
2 Preliminaries
The image of the convex body K at time \(t\geq0\) under the normal flow having speed \({p_{W}}(u)\) (the Wulff flow associated to W) is \(K+tW\).
Proposition 1
Poincaré lemma [1]
3 Bonnesen-style Wulff isoperimetric inequalities
To prove our main results, we need the following lemmas.
Lemma 1
Let K, W be two convex bodies in \(\mathbb{R}^{2}\). Let \(p_{K}(\theta)\) and \(p_{W}(\theta)\) be support functions of K and W, respectively. If \(W\subseteq K\) such that \(p_{W}(\theta)\ne p_{K}(\theta)\), \(\theta\in[\theta_{0},\theta_{0}+\pi]\) for some \(\theta_{0}\), then there exist \(\epsilon >0\) and \(v\in S^{1}\) such that \(W+\epsilon\cdot v\subset K\).
Proof
Lemma 2
Proof
There is at least one point where \(\partial(r_{W}W)\) is tangent to ∂K for \(\theta\in[\theta_{0}, \theta_{0}+\pi]\) with all \(\theta_{0}\). If the conclusion fails, that is, there exists \(\theta_{0}\) such that \(p_{r_{W}W}(\theta) \ne p_{K}(\theta) \) for \(\theta\in[\theta_{0}, \theta_{0}+\pi]\), choose the vector v corresponding to the angle \(\theta_{0}+\frac{\pi}{2}\). By Lemma 1, if we move \(r_{W}W\) by v for \(\epsilon>0\) small enough, then \(r_{W}W+\epsilon\cdot v\) continues to lie in the interior of K and has no points of tangency. This contradicts the maximality of \(r_{W}\).
Remark
Inequality (26) has been mentioned in Green and Osher’s work (cf. [1]) without proof. For general convex bodies, Luo, Xu and Zhou [17] have also obtained inequality (26) by the integral geometry method. However, it is difficult to obtain the equality condition of inequality (26) for general convex bodies. Via the method of convex geometric analysis, a complete proof of inequality (26) with equality condition is given in [9].
By (28) or (29), the sufficient condition for root existence of equation \(A_{K, W}(t)=0\) is that the discriminant of \(A_{K, W}(t)=0\) is non-negative. We obtain the following Wulff isoperimetric inequality.
Corollary 1
Proof of Theorem 1
Let W be the unit disc, then \(L_{K,W}^{2}=L_{K}^{2}\), \(A_{W}=\pi\). Therefore we have the following.
Corollary 2
It should be noted that (32) is obtained in [24], which is stronger than the Bonnesen isoperimetric inequality (4).
4 Reverse Bonnesen-style Wulff isoperimetric inequalities
To prove reverse Bonnesen-style Wulff isoperimetric inequalities in Theorem 2, we need the following Wulff isoperimetric inequalities.
Lemma 3
Proof
Proof of Theorem 2
Let W be a unit disc. Direct consequences of Theorem 2 are as follows.
Corollary 3
The reverse Bonnesen-style inequality (42) is obtained by Bokowski, Heil, Zhou, Ma and Xu (cf. [4, 27]).
Declarations
Acknowledgements
The authors would like to thank anonymous referees for helpful comments and suggestions that directly led to the improvement of the original manuscript. The corresponding author is supported in part by Natural Science Foundation Project (grant number: # 11671325).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Green, M, Osher, S: Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves. Asian J. Math. 3, 659-676 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Banchoff, TF, Pohl, WF: A generalization of the isoperimetric inequality. J. Differ. Geom. 6, 175-213 (1971) MathSciNetMATHGoogle Scholar
- Berger, M: Geometry I. Springer, Berlin (1989) Google Scholar
- Bokowski, J, Heil, E: Integral representations of quermassintegrals and Bonnesen-style inequalities. Arch. Math. 47, 79-89 (1986) MathSciNetView ArticleMATHGoogle Scholar
- Bonnesen, T: Les Problèmes des Isopérimètres et des Isépiphanes. Gauthie-Villars, Paris (1929) MATHGoogle Scholar
- Bonnesen, T: Über eine Verschärfung der isoperimetrischen Ungleichheit des Kreises in der Ebene und auf der Kugeloberfläche nebst einer Anwendung auf eine Minkowskische Ungleichheit für konvexe Körper. Math. Ann. 84, 216-227 (1921) MathSciNetView ArticleMATHGoogle Scholar
- Bonnesen, T: Über das isoperimetrische Defizit ebener Figuren. Math. Ann. 91, 252-268 (1924) MathSciNetView ArticleMATHGoogle Scholar
- Bonnesen, T: Quelques problèms isopérimetriques. Acta Math. 48, 123-178 (1926) View ArticleMATHGoogle Scholar
- Böröczky, K, Lutwak, E, Yang, D, Zhang, G: The log-Brunn-Minkowski inequality. Adv. Math. 231, 1974-1997 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Burago, YD, Zalgaller, VA: Geometric Inequalities. Springer, Berlin (1988) View ArticleMATHGoogle Scholar
- Enomoto, K: A generalization of the isoperimetric inequality on \(S^{2}\) and flat tori in \(S^{3}\). Proc. Am. Math. Soc. 120, 553-558 (1994) MathSciNetMATHGoogle Scholar
- Goldstein, T, Bresson, X, Osher, S: Global minimization of Markov random fields with applications to optical flow. Inverse Probl. Imaging 6, 623-644 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Grinberg, E, Ren, D, Zhou, J: The symmetric isoperimetric deficit and the containment problem in a plan of constant curvature. Preprint Google Scholar
- Grinberg, E, Li, S, Zhang, G, Zhou, J: Integral Geometry and Convexity. World Scientific, Singapore (2006) View ArticleGoogle Scholar
- Grinberg, E: Isoperimetric inequalities and identities for k-dimensional cross-sections of convex bodies. Math. Ann. 291, 75-86 (1991) MathSciNetView ArticleMATHGoogle Scholar
- Grinberg, E, Zhang, G: Convolutions, transforms, and convex bodies. Proc. Lond. Math. Soc. 78, 77-115 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Luo, M, Xu, W, Zhou, J: Translative containment measure and symmetric mixed isohomothetic inequalities. Sci. China Math. 58, 2593-2610 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Oberman, A, Osher, S, Takei, R, Tsai, R: Numerical methods for anisotropic mean curvature flow based on a discrete time variational formulation. Commun. Math. Sci. 9, 637-662 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Osserman, R: The isoperimetric inequality. Bull. Am. Math. Soc. 84, 1182-1238 (1978) MathSciNetView ArticleMATHGoogle Scholar
- Osserman, R: Bonnesen-style isoperimetric inequalities. Am. Math. Mon. 86, 1-29 (1979) MathSciNetView ArticleMATHGoogle Scholar
- Ren, D: Topics in Integral Geometry. World Scientific, Singapore (1994) MATHGoogle Scholar
- Santaló, LA: Integral Geometry and Geometric Probability. Addison-Wesley, Reading (1976) MATHGoogle Scholar
- Schneider, R: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (2014) MATHGoogle Scholar
- Zeng, C, Zhou, J, Yue, S: The symmetric mixed isoperimetric inequality of two planar convex domains. Acta Math. Sin., Chinese Ser. 55, 355-362 (2012) MathSciNetMATHGoogle Scholar
- Zhou, J: On Bonnesen-style inequalities. Acta Math. Sin. 50, 1397-1402 (2007) MATHGoogle Scholar
- Zhou, J, Chen, F: The Bonnesen-type inequalities in a plane of constant curvature. J. Korean Math. Soc. 44, 1-10 (2007) MathSciNetMATHGoogle Scholar
- Zhou, J, Ma, L, Xu, W: On the isoperimetric deficit upper limit. Bull. Korean Math. Soc. 50, 175-184 (2013) MathSciNetView ArticleMATHGoogle Scholar