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Inequalities for the Mixed Radial Blaschke-Minkowski Homomorphisms and the Applications

Received: 08 December 2019    Accepted: 19 December 2019    Published: 17 April 2020
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Abstract

The notion of intersection body is introduced by Lutwak in 1988, it is one of important research contents and led to the studies of Busemann-Petty problem in the Brunn-Minkowski theory. Based on the properties of the intersection bodies, Schuster introduced the notion of radial Blaschke-Minkowski homomorphisms and proved a lot of related inequalities. In this paper, by applying the dual mixed volume theory and analytic inequalities, we first give a lower bound of the dual quermassintegrals for the mixed radial Blaschke-Minkowski homomorphisms. As its an application, we get a reverse form of the well-known Busemann intersection inequality. Further, a Brunn-Minkowski type inequality of the Lp radial Minkowski sum for the dual quermassintegrals of mixed radial Blaschke-Minkowski homomorphisms is established, and then the intersection body version of this Brunn-Minkowski type inequality is yielded. From this, we not only extend Schuster's related result but also obtain the Brunn-Minkowski type inequalities of Lp harmonic radial sum and Lp radial Blaschke sum, respectively.

DOI 10.11648/j.acm.20200901.12
Published in Applied and Computational Mathematics (Volume 9, Issue 1, February 2020)
Page(s) 14-19
Creative Commons

This is an Open Access article, 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 or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Dual Quermassintegral, Intersection Body, Radial Blaschke-Minkowski Homomorphism, Busemann Intersection Inequality, Lp Radial Minkowski Sum

References
[1] R. J. Gardner, ``Geometric Tomography", Second ed., Cambridge Univ. Press, Cambridge, 2006.
[2] E. Lutwak, ``intersection bodies and dual mixed volumes", Adv. Math., 71 (1988), 531-538.
[3] R. J. Gardner, ``Intersection bodies and the Busemann-Petty problem", Trans. Amer. Math. Soc., 342 (1994) 435-445.
[4] A. Koldlbsky, ``A functional analytic approach to intersection bodies", Geom. Funct. Anal., 10 (2000), 1507-1526.
[5] G. S. Leng and C. J. Zhao, ``Inequalities for dual quermassintegrals of mixed intersction bodies", Proc. Indian Acad. Sci., 115 (2003), 79-91.
[6] G. S. Leng and C. J. Zhao, ``Brunn-Minkowski inequality for mixed intersection bodies", J. Math. Anal. Appl., 301 (2005), 115-123.
[7] R. Schneider, ``Convex Bodies: The Brunn-Minkowski theory", 2nd edn, Cambridge University Press, Cambridge, 2014.
[8] G. Y. Zhang, ``Centered bodies and dual mixed volumes", Trans. Amer. Math. Soc., 345 (1994), 777-801.
[9] G. Y. Zhang, ``A positive solution to the Busemann-Petty problem in $R^4$", Ann. of Math., 149 (1999), 2: 535-543.
[10] F. E. Schuster, ``Volume inequalities and additive maps of convex bodies", Mathematika, 53 (2006), 211-234.
[11] F. E. Schuster, ``Valuations and Busemann-Petty type problems", Adv. Math., 219 (2008), 344-368.
[12] W. Wang, L. J. Liu and B. W. He ``Lp radial Blaschke-Minkowski homomorphisms", Taiwan J. Math., 15 (2011), 1183-1199.
[13] B. Chen and W. D. Wang, ``A type of Busemann-Petty problems for Blaschke-Minkowski homomorphisms", Wuhan University Journal of Natural Sciences, 23 (2018), 289-294.
[14] B. Chen and W. D. Wang, ``Some inequalities for Lp radial Blaschke-Minkowski homomorphisms", Quaest. Math., 42 (2019), 391-405.
[15] Y. B. Feng, W. D. Wang and J. Yuan, ``Differences of quermass- and dual quermassintegrals of Blaschke-Minkowski and radial Blaschke-Minkowski homomorphisms", B. Belg. Math. Soc-Sim, 21 (2014), 577-592.
[16] L. J. Liu, ``Mixed radial Blaschke-Minkowski homomorphisms and comparison of volumes", Math. Inequal. Appl., 16 (2013), 401-412.
[17] Z. H. Shen and W. D. Wang, ``Lp radial Blaschke-Minkowski homomorphisms and Lp dual affine surface areas", Mathematics, 7 (2019), 343: 14 pages.
[18] W. D. Wang, H. P. Chen and Y. Y. Zhang, ``Busemann-Petty problem for the $i$-th radial Blaschke-Minkowski homomorphisms", Filomat, 32 (2018), 6819-6827.
[19] B. Wei, W. D. Wang and F. H. Lu, ``Inequalities for radial Blaschke-Minkowski homomorphisms", Ann. Pol. Math., 113 (2015), 243-253.
[20] C. J. Zhao, ``Radial Blaschke-Minkowski homomorphisms and volume differences", Geom. Dedicata, 154 (2011), 81-91.
[21] C. J. Zhao, ``On radial Blaschke-Minkowski homomorphisms", Geom. Dedicata, 167 (2013), 1-10.
[22] C. J. Zhao, ``On raidal and polar Blaschke-Minkowski homomorphisms", Proc. Amer. Math. Soc., 141 (2013), 667-676.
[23] C. J. Zhao, ``On Blaschke-Minkowski homomorphisms and radial Blaschke-Minkowski homomorphisms", J. Geom. Anal., 25 (2015), 1-16.
[24] X. Zhao and W. D. Wang, ``Brunn-Minkowski inequalities for the Lp and Lp radial Blaschke-Minkowski homomorphisms", IAENG International Journal of Applied Mathematics, 49 (2019), 1-5.
[25] Y. Zhou and W. D. Wang, ``Some Brunn-Minkowski type inequalities for Lp radial Blaschke-Minkowski homomorphisms", J. Inequal. Appl., 2016 (2016), 183, 11 pages.
[26] H. Busemann, ``Volum in terms of concurrent cross-sections", Pacofoc J. Math., 3 (1953), 1-12.
[27] R. J. Gardner, ``The Brunn-Minkowski inequality", Bull. Amer. Math. Soc., 39 (2002), 355-405.
[28] G. Y. Zhang, ``Convolutions, transforms, and convex bodies", Proc. London Math. Soc., 78 (1999), 77-115.
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Author Information
  • Department of Mathematics, China Three Gorges University, Yichang, China; Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, China

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    Weidong Wang. (2020). Inequalities for the Mixed Radial Blaschke-Minkowski Homomorphisms and the Applications. Applied and Computational Mathematics, 9(1), 14-19. https://doi.org/10.11648/j.acm.20200901.12

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    Weidong Wang. Inequalities for the Mixed Radial Blaschke-Minkowski Homomorphisms and the Applications. Appl. Comput. Math. 2020, 9(1), 14-19. doi: 10.11648/j.acm.20200901.12

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    Weidong Wang. Inequalities for the Mixed Radial Blaschke-Minkowski Homomorphisms and the Applications. Appl Comput Math. 2020;9(1):14-19. doi: 10.11648/j.acm.20200901.12

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  • @article{10.11648/j.acm.20200901.12,
      author = {Weidong Wang},
      title = {Inequalities for the Mixed Radial Blaschke-Minkowski Homomorphisms and the Applications},
      journal = {Applied and Computational Mathematics},
      volume = {9},
      number = {1},
      pages = {14-19},
      doi = {10.11648/j.acm.20200901.12},
      url = {https://doi.org/10.11648/j.acm.20200901.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20200901.12},
      abstract = {The notion of intersection body is introduced by Lutwak in 1988, it is one of important research contents and led to the studies of Busemann-Petty problem in the Brunn-Minkowski theory. Based on the properties of the intersection bodies, Schuster introduced the notion of radial Blaschke-Minkowski homomorphisms and proved a lot of related inequalities. In this paper, by applying the dual mixed volume theory and analytic inequalities, we first give a lower bound of the dual quermassintegrals for the mixed radial Blaschke-Minkowski homomorphisms. As its an application, we get a reverse form of the well-known Busemann intersection inequality. Further, a Brunn-Minkowski type inequality of the Lp radial Minkowski sum for the dual quermassintegrals of mixed radial Blaschke-Minkowski homomorphisms is established, and then the intersection body version of this Brunn-Minkowski type inequality is yielded. From this, we not only extend Schuster's related result but also obtain the Brunn-Minkowski type inequalities of Lp harmonic radial sum and Lp radial Blaschke sum, respectively.},
     year = {2020}
    }
    

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  • TY  - JOUR
    T1  - Inequalities for the Mixed Radial Blaschke-Minkowski Homomorphisms and the Applications
    AU  - Weidong Wang
    Y1  - 2020/04/17
    PY  - 2020
    N1  - https://doi.org/10.11648/j.acm.20200901.12
    DO  - 10.11648/j.acm.20200901.12
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    JO  - Applied and Computational Mathematics
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    UR  - https://doi.org/10.11648/j.acm.20200901.12
    AB  - The notion of intersection body is introduced by Lutwak in 1988, it is one of important research contents and led to the studies of Busemann-Petty problem in the Brunn-Minkowski theory. Based on the properties of the intersection bodies, Schuster introduced the notion of radial Blaschke-Minkowski homomorphisms and proved a lot of related inequalities. In this paper, by applying the dual mixed volume theory and analytic inequalities, we first give a lower bound of the dual quermassintegrals for the mixed radial Blaschke-Minkowski homomorphisms. As its an application, we get a reverse form of the well-known Busemann intersection inequality. Further, a Brunn-Minkowski type inequality of the Lp radial Minkowski sum for the dual quermassintegrals of mixed radial Blaschke-Minkowski homomorphisms is established, and then the intersection body version of this Brunn-Minkowski type inequality is yielded. From this, we not only extend Schuster's related result but also obtain the Brunn-Minkowski type inequalities of Lp harmonic radial sum and Lp radial Blaschke sum, respectively.
    VL  - 9
    IS  - 1
    ER  - 

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