New inequalities for star bodies
- Yanxiong Yan^{1},
- Liangcai Zhang^{2} and
- Min Zhou^{3}Email author
DOI: 10.1186/s13660-016-1081-5
© Yan et al. 2016
Received: 17 November 2015
Accepted: 9 May 2016
Published: 1 June 2016
Abstract
In this paper, we investigate the radial addition and Blaschke addition and get some new Brunn-Minkowski inequalities associated with dual quermassintegrals and chord integral for star bodies.
Keywords
Brunn-Minkowski theory radial addition dual Brunn-Minkowski theory mixed volumes Blaschke additionMSC
52A20 52A401 Introduction
The Brunn-Minkowski theory, which is the so-called mixed-volume theory, is the classical core of the geometry of convex bodies. This theory originated from the thesis of Hermann Brunn in 1887, and in its essential part is the creation of Hermann Minkowski around the turn of the century. The well-known survey of Bonnesen and Fenchel collected an impressive body of results in 1934, though important developments, by the work of Aleksandrov and others in the 1930s, were still to come. In recent years, the theory of convex bodies was expanded considerably, new topics have been developed rapidly, and originally neglected branches of the subject have gained in interest. For example, the Brunn-Minkowski theory has remained of constant interest owing to its various new applications and connections with other fields.
During the last three decades, the Brunn-Minkowski theory has achieved important developments. In the 1970s, Lutwak’s dual Brunn-Minkowski theory had come out, which helped to achieve major breakthrough of solving the Busemann-Petty problem in the 1990s. In the dual theory, compared with the Brunn-Minkowski theory, convex bodies are replaced by star-shaped bodies, and projections onto subspaces are replaced by intersections with subspaces. The machinery of the dual theory includes dual mixed volumes and intersection bodies (see [1, 2, 4–12]).
The radial addition and Blaschke addition are still playing a crucial role in the Brunn-Minkowski theory. In this article, we continue to investigate the radial addition and Blaschke addition and get some new Brunn-Minkowski inequalities associated with dual quermassintegrals and chord integral for star bodies.
2 Preliminaries
A set K of points in the Euclidean space \(\mathbb{R}^{n}\) is convex if for any \(x,y\in K\), we have \(0\leq\lambda\leq1\) and \(\lambda x+(1-\lambda)y\in K\). A domain is a set with nonempty interiors. A convex body is a compact convex domain. The set of convex bodies in \(\mathbb{R}^{n}\) is denoted by \(\mathcal{K}^{n}\). Let \(\mathcal{K}_{o}^{n}\) be the class of members of \(\mathcal{K}^{n}\) containing the origin in their interiors. We write V for the n-dimensional Lebesgue measure and \(\mathcal{H}^{n-1}\) for the \((n-1)\)-dimensional Hausdorff measure. We denote by \(S^{n-1}\) the surface of the unit ball in \(\mathbb{R}^{n}\).
By the dual Minkowski inequality we can obtain the dual Brunn-Minkowski inequality (see [10]):
We also need the following Minkowski inequality for integrals is needed (see [13]):
If \(p<0\) or \(0< p<1\), the inequality is reverse.
3 Inequalities of dual quermassintegrals
Theorem 1
For \(M,N\in\mathcal{S}_{o}^{n}\), we have \(\tilde{W}_{2n}(M)^{\frac{1}{n}}+\tilde{W}_{2n}(N)^{\frac{1}{n}}\geq 4\tilde{W}_{2n}(M\,\tilde{+}\,N)^{\frac{1}{n}}\) with equality if and only if M and N are the same.
Proof
For Blaschke addition, we have the following theorem.
Theorem 2
Proof
Similarly, for \(i<1\), we can get that the reverse inequality of Minkowski’s inequality for integrals.
Particularly, if \(i=0\), then Theorem 2 implies the following: □
Corollary 3.1
4 Inequalities of chord integral of the star body
Theorem 3
Proof
Theorem 4
Proof
Similarly, we can prove the case of \(i< n-1\) with the reverse inequality, which follows by the Minkowski’s inequality for integrals. □
Particularly, if \(i=2n\), then by Theorem 4 we have the following corollary.
Corollary 4.1
Theorem 5
Proof
Similarly, if \(i<1\), then we can get that the reverse inequality, which follows by the Minkowski inequality for integrals. □
Declarations
Acknowledgements
Authors would like to thank two anonymous referees for many helpful comments and suggestions that directly led to the improvement of the original manuscript. Supported by Natural Science Foundation of China (Grant Nos. 11171364; 11271301; 11471266; 11426182); by ‘Fundamental Research Funds for the Central Universities’ (Grant Nos. XDJK2016B037; SWU115052; SWU1509190); by Natural Science Foundation Project of CQ CSTC (Grant Nos. cstc2014jcyjA00010; 2010BB9206) and by the Project-sponsored by SRF for ROCS, SEM.
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
- Gardner, R: On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies. Bull. Am. Math. Soc. 30, 222-226 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Burago, YD, Zalgaller, VA: Geometric Inequalities. Springer, Berlin (1988) View ArticleMATHGoogle Scholar
- Schneider, R: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (2014) MATHGoogle Scholar
- Gardner, R: Geometric Tomography, 2nd edn. Cambridge University Press, New York (2006) View ArticleMATHGoogle Scholar
- Gardner, R: Intersection bodies and the Busemann-Petty problem. Trans. Am. Math. Soc. 342, 435-445 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Gardner, R: A positive answer to the Busemann-Petty problem in three dimensions. Ann. Math. 140, 435-447 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Goodey, P, Weil, W: Intersection bodies and ellipsoids. Mathematika 42, 295-304 (1995) MathSciNetView ArticleMATHGoogle Scholar
- Klain, D: Star Measures and Dual Mixed Volumes. Ph.D. thesis, MIT, Cambridge (1994)
- Klain, D: Star valuations and dual mixed volumes. Adv. Math. 121, 80-101 (1996) MathSciNetView ArticleMATHGoogle Scholar
- Lutwak, E: Intersection bodies and dual mixed volumes. Adv. Math. 71, 232-261 (1988) MathSciNetView ArticleMATHGoogle Scholar
- Lutwak, E: Centroid bodies and dual mixed volumes. Proc. Lond. Math. Soc. 60, 365-391 (1990) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, G: Centered bodies and dual mixed volumes. Trans. Am. Math. Soc. 345, 777-801 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, London (1934) MATHGoogle Scholar