General \(L_{p}\)-mixed-brightness integrals
- Li Yan^{1} and
- Weidong Wang^{1}Email author
https://doi.org/10.1186/s13660-015-0708-2
© Yan and Wang 2015
Received: 31 December 2014
Accepted: 21 May 2015
Published: 12 June 2015
Abstract
The notion of mixed-brightness integrals was introduced by Li and Zhu. In this paper, motivated by the notion of general \(L_{p}\)-projection bodies, introduced by Haberl and Schuster, we define general \(L_{p}\)-mixed-brightness integrals and determine their extremal values, as well as several other inequalities for them.
Keywords
MSC
1 Introduction
Let \(\mathcal{ K}^{n}\) denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space \(\mathbb {R}^{n}\). For the set of convex bodies containing the origin in their interiors and the set of origin-symmetric convex bodies in \(\mathbb {R}^{n}\), we write \(\mathcal{ K}^{n}_{o}\) and \(\mathcal{ K}^{n}_{os}\), respectively. Let \(\mathcal{ S}^{n}_{o}\) denote the set of star bodies (about the origin) in \(\mathbb{R}^{n}\) and let \(S^{n-1}\) denote the unit sphere in \(\mathbb {R}^{n}\). By \(V(K)\) we denote the n-dimensional volume of a body K and for the standard unit ball B in \(\mathbb{R}^{n}\), we write \(\omega _{n}\) for its volume.
Further, Li and Zhu [3] established the following Fenchel-Aleksandrov type inequality for mixed-brightness integrals.
Theorem 1.A
More recently, Zhou et al. [4] obtained Brunn-Minkowski type inequalities for mixed-brightness integrals.
For general \(L_{p}\)-projection bodies, Haberl and Schuster [16] proved the general \(L_{p}\)-Petty projection inequality and determined the extremal values of volume for polars of general \(L_{p}\)-projection bodies. Wang and Wan [17] investigated Shephard type problems for general \(L_{p}\)-projection bodies. Wang and Feng [18] established general \(L_{p}\)-Petty affine projection inequality. These investigations were the starting point of a new and rapidly evolving asymmetric \(L_{p}\)-Brunn-Minkowski theory (see [13–32]).
Remark 1.1
In this paper, we establish several inequalities for general \(L_{p}\)-mixed-brightness integrals. First, we determine the extremal values of general \(L_{p}\)-mixed-brightness integrals.
Theorem 1.1
Next, we obtain a Brunn-Minkowski type inequality for general \(L_{p}\)-mixed-brightness integrals.
Theorem 1.2
Here, \(\lambda\circ K\oplus_{p} \mu\circ L\) denotes the \(L_{p}\)-Blaschke combination of K and L.
Next, we extend inequality (1.2) to general \(L_{p}\)-mixed-brightness integrals.
Theorem 1.3
Taking \(m=n\) in Theorem 1.3 and using (1.10), we obtain the following corollary.
Corollary 1.1
Moreover, we also establish the following cyclic inequality for general \(L_{p}\)-mixed-brightness integrals.
Theorem 1.4
Taking \(i=0\), \(k=n\) in Theorem 1.4 and using (1.11), we obtain the following result.
Corollary 1.2
Let \(L=B\) in Theorem 1.4, we also have the following result.
Corollary 1.3
2 Notation and background material
2.1 Radial function and polars of convex bodies
2.2 \(L_{p}\)-combinations of convex and star bodies
In [37] Wang and Leng established the following Brunn-Minkowski type inequality for dual quermassintegrals with respect to an \(L_{p}\)-harmonic radial combination of star bodies.
Theorem 2.A
2.3 General \(L_{p}\)-projection bodies
Moreover, they [16] determined the following extremal values of the volume for polars of general \(L_{p}\)-projection bodies.
Theorem 2.B
Here, \(\Pi^{\tau,\ast}_{p}K\) denotes the polar of the general \(L_{p}\)-projection body \(\Pi^{\tau}_{p}K\).
3 Proofs of the main theorems
In this section, we will prove Theorems 1.1-1.3.
To complete the proofs of Theorems 1.1-1.2, we require the following a lemma.
Lemma 3.1
Proof
Proof of Theorem 1.1
According to the equality conditions of inequality (2.11), we know that if K is not origin-symmetric and p is not an odd integer, there is equality in the left inequality of (1.12) if and only if \(\tau= 0\) and equality in the right inequality of (1.12) if and only if \(\tau= \pm 1\). □
Proof of Theorem 1.2
From the equality conditions of inequality (2.9), we see that equality holds in (1.13) if and only if \(\Pi_{p}^{\tau,\ast}K\) and \(\Pi _{p}^{\tau,\ast}L\) are dilates, i.e., \(\Pi_{p}^{\tau}K\) and \(\Pi_{p}^{\tau}L\) are dilates. This means equality holds in (1.13) if and only if K and L have similar general \(L_{p}\)-brightness.
Similarly, if \(n-p< i< n\) or \(i>n\), then \(2n-i< n\) or \(n<2n-i<n+p\). Thus, using (3.1), (3.3), and inequality (2.8), we obtain inequality (1.14).
If \(i=n-p\), then \(2n-i=n+p\). This combined with Theorem 2.A, shows that equality always holds in (1.13) or (1.14). □
The proof of Theorem 1.3 requires the following inequality [3].
Lemma 3.2
Proof of Theorem 1.3
Proof of Theorem 1.4
Declarations
Acknowledgements
The authors would like to sincerely thank the referees for all valuable and helpful comments and suggestions which made the paper more accurate and readable. Research is supported in part by the Natural Science Foundation of China (Grant No. 11371224) and Foundation of Degree Dissertation of Master of China Three Gorges University (Grant No. 2015PV070).
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, RJ: Geometric Tomography, 2nd edn. Cambridge University Press, Cambridge (2006) MATHView ArticleGoogle Scholar
- Schneider, R: Convex Bodies: The Brunn-Minkowski Theory, 2nd edn. Cambridge University Press, Cambridge (2014) Google Scholar
- Li, N, Zhu, BC: Mixed brightness-integrals of convex bodies. J. Korean Math. Soc. 47(5), 935-945 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Zhou, YP, Wang, WD, Feng, YB: The Brunn-Minkowski type inequalities for mixed brightness-integrals. Wuhan Univ. J. Nat. Sci. 19(4), 277-282 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Lutwak, E, Yang, D, Zhang, GY: \(L_{p}\) affine isoperimetric inequalities. J. Differ. Geom. 56, 111-132 (2000) MATHMathSciNetGoogle Scholar
- Abardia, J, Bernig, A: Projection bodies in complex vector spaces. Adv. Math. 227, 830-846 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Abardia, J: Difference bodies in complex vector spaces. J. Funct. Anal. 263, 3588-3603 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Abardia, J: Minkowski valuations in a 2-dimensional complex vector space. Int. Math. Res. Not. 2015, 1247-1262 (2015) MATHGoogle Scholar
- Haberl, C: Minkowski valuations intertwining with the special linear group. J. Eur. Math. Soc. 14, 1565-1597 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Parapatits, L, Schuster, FE: The Steiner formula for Minkowski valuations. Adv. Math. 230, 978-994 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Parapatits, L, Wannerer, T: On the inverse Klein map. Duke Math. J. 162, 1895-1922 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Schuster, FE: Crofton measures and Minkowski valuations. Duke Math. J. 154, 1-30 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Schuster, FE, Wannerer, T: Even Minkowski valuations. Am. J. Math. (in press) Google Scholar
- Schuster, FE, Weberndorfer, M: Volume inequalities for asymmetric Wulff shapes. J. Differ. Geom. 92, 263-283 (2012) MATHMathSciNetGoogle Scholar
- Ludwig, M: Minkowski valuations. Trans. Am. Math. Soc. 357, 4191-4213 (2005) MATHMathSciNetView ArticleGoogle Scholar
- Haberl, C, Schuster, F: General \(L_{p}\)-affine isoperimetric inequalities. J. Differ. Geom. 83, 1-26 (2009) MATHMathSciNetGoogle Scholar
- Wang, WD, Wan, XY: Shephard type problems for general \(L_{p}\)-projection bodies. Taiwan. J. Math. 16(5), 1749-1762 (2012) MATHMathSciNetGoogle Scholar
- Wang, WD, Feng, YB: A general \(L_{p}\)-version of Petty’s affine projection inequality. Taiwan. J. Math. 17(2), 517-528 (2013) MATHView ArticleGoogle Scholar
- Feng, YB, Wang, WD: General \(L_{p}\)-harmonic Blaschke bodies. Proc. Indian Acad. Sci. Math. Sci. 124(1), 109-119 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Feng, YB, Wang, WD, Lu, FH: Some inequalities on general \(L_{p}\)-centroid bodies. Math. Inequal. Appl. 18(1), 39-49 (2015) MATHMathSciNetGoogle Scholar
- Haberl, C: \(L_{p}\)-Intersection bodies. Adv. Math. 4, 2599-2624 (2008) MathSciNetView ArticleGoogle Scholar
- Haberl, C, Ludwig, M: A characterization of \(L_{p}\) intersection bodies. Int. Math. Res. Not. 2006, Art ID 10548 (2006) MathSciNetGoogle Scholar
- Haberl, C, Schuster, FE: Asymmetric affine \(L_{p}\) Sobolev inequalities. J. Funct. Anal. 257, 641-658 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Haberl, C, Schuster, FE, Xiao, J: An asymmetric affine Pólya-Szegö principle. Math. Ann. 352, 517-542 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Ludwig, M: Intersection bodies and valuations. Am. J. Math. 128, 1409-1428 (2006) MATHMathSciNetView ArticleGoogle Scholar
- Parapatits, L: \(\operatorname{SL}(n)\)-Covariant \(L_{p}\)-Minkowski valuations. J. Lond. Math. Soc. 89, 397-414 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Parapatits, L: \(\operatorname{SL}(n)\)-Contravariant \(L_{p}\)-Minkowski valuations. Trans. Am. Math. Soc. 366, 1195-1211 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Schuster, FE, Wannerer, T: \(\operatorname{GL}(n)\) contravariant Minkowski valuations. Trans. Am. Math. Soc. 364, 815-826 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Wang, WD, Li, YN: Busemann-Petty problems for general \(L_{p}\)-intersection bodies. Acta Math. Sin. Engl. Ser. 31(5), 777-786 (2015) MathSciNetView ArticleGoogle Scholar
- Wang, WD, Ma, TY: Asymmetric \(L_{p}\)-difference bodies. Proc. Am. Math. Soc. 142(7), 2517-2527 (2014) MATHView ArticleGoogle Scholar
- Wannerer, T: \(\operatorname{GL}(n)\) equivariant Minkowski valuations. Indiana Univ. Math. J. 60, 1655-1672 (2011) MATHMathSciNetView ArticleGoogle Scholar
- Weberndorfer, M: Shadow systems of asymmetric \(L_{p}\) zonotopes. Adv. Math. 240, 613-635 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Lutwak, E: Dual mixed volumes. Pac. J. Math. 58, 531-538 (1975) MATHMathSciNetView ArticleGoogle Scholar
- Firey, WJ: p-Means of convex bodies. Math. Scand. 10, 17-24 (1962) MATHMathSciNetGoogle Scholar
- Lutwak, E: The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131-150 (1993) MATHMathSciNetGoogle Scholar
- Lutwak, E: The Brunn-Minkowski-Firey theory II: affine and geominimal surface areas. Adv. Math. 118, 244-294 (1996) MATHMathSciNetView ArticleGoogle Scholar
- Wang, WD, Leng, GS: A correction to our paper ‘\(L_{p}\)-dual mixed quermassintegrals’. Indian J. Pure Appl. Math. 38(6), 609 (2007) MathSciNetGoogle Scholar
- Besau, F, Schuster, FE: Binary operations in spherical convex geometry. arXiv:1407.1153
- Gardner, RJ, Hug, D, Weil, W: Operations between sets in geometry. J. Eur. Math. Soc. 15, 2297-2352 (2013) MATHMathSciNetView ArticleGoogle Scholar
- Gardner, RJ, Hug, D, Weil, W: The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities. J. Differ. Geom. 97, 427-476 (2014) MATHMathSciNetGoogle Scholar
- Gardner, RJ, Parapatits, L, Schuster, FE: A characterization of Blaschke addition. Adv. Math. 254, 396-418 (2014) MATHMathSciNetView ArticleGoogle Scholar
- Li, J, Yuan, S, Leng, G: Lp-Blaschke valuations. Trans. Am. Math. Soc. 367, 3161-3187 (2015) MathSciNetView ArticleGoogle Scholar