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

Weidong Wang

doi:doi:10.11648/j.acm.20200901.12

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

Weidong Wang

Published in Applied and Computational Mathematics (Volume 9, Issue 1)

Received: 8 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 L_p 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 L_p harmonic radial sum and L_p radial Blaschke sum, respectively.

Published in	Applied and Computational Mathematics (Volume 9, Issue 1)
DOI	10.11648/j.acm.20200901.12
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), 2020. Published by Science Publishing Group

Keywords

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

References

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[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 ``L_p 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 L_p 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, ``L_p radial Blaschke-Minkowski homomorphisms and L_p 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 L_p and L_p radial Blaschke-Minkowski homomorphisms", IAENG International Journal of Applied Mathematics, 49 (2019), 1-5.
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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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ACS Style

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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AMA Style

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://article.sciencepublishinggroup.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
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 14
EP  - 19
PB  - Science Publishing Group
SN  - 2328-5613
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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Author Information

Weidong Wang

Department of Mathematics, 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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ACS Style

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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AMA Style

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

Copy | Download

@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://article.sciencepublishinggroup.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}
}

Copy | Download

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
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 14
EP  - 19
PB  - Science Publishing Group
SN  - 2328-5613
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  -

Copy | Download