On Brunn-Minkowski type inequality
Volume 11, Issue 6, pp 762--769
http://dx.doi.org/10.22436/jnsa.011.06.03
Publication Date: April 18, 2018
Submission Date: December 01, 2016
Revision Date: November 15, 2017
Accteptance Date: March 01, 2018
-
2611
Downloads
-
4942
Views
Authors
Lewen Ji
- Department of Mathematics, East China University of Technology, Nanchang 330013, China.
- Department of Mathematics, Shanghai University, Shanghai 200444,, China.
Zhenbing Zeng
- Department of Mathematics, Shanghai University, Shanghai 200444,, China.
Jingjing Zhong
- School of Public Finance and Public Administration, Jiangxi University of Finance and Economics, Nanchang 330013, China.
Abstract
The notion of Aleksandrov body in the classical Brunn-Minkowski theory is extended to that
of Orlicz-Aleksandrov body in the Orlicz Brunn-Minkowski theory. The analogs of the Brunn-Minkowski type inequality and the first variations of volume are established via Orlicz-Aleksandrov body. We also make some considerations for the polar of Orlicz combination.
Share and Cite
ISRP Style
Lewen Ji, Zhenbing Zeng, Jingjing Zhong, On Brunn-Minkowski type inequality, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 6, 762--769
AMA Style
Ji Lewen, Zeng Zhenbing, Zhong Jingjing, On Brunn-Minkowski type inequality. J. Nonlinear Sci. Appl. (2018); 11(6):762--769
Chicago/Turabian Style
Ji, Lewen, Zeng, Zhenbing, Zhong, Jingjing. "On Brunn-Minkowski type inequality." Journal of Nonlinear Sciences and Applications, 11, no. 6 (2018): 762--769
Keywords
- Orlicz-Aleksandrov body
- Brunn-Minkowski type inequality
- Orlicz combination
MSC
References
-
[1]
A. D. Aleksandrov, On the theory of mixed volumes.I. Extension of certain concepts in the theory of convex bodies, Mat. Sb., 2 (1937), 947–972.
-
[2]
K. J. Böröczky, E. Lutwak, D. Yang, G. Zhang , The log-Brunn-Minkowski inequality, Adv. Math., 231 (2012), 1974– 1997.
-
[3]
S. Campi, P. Gronchi , The Lp Busemann-Petty centroid inequality, Adv. Math., 167 (2002), 128–141.
-
[4]
S. Campi, P. Gronchi , On volume product inequalities for convex sets , Proc. Amer. Math. Soc., 134 (2006), 2393–2402.
-
[5]
W. J. Firey , Mean cross-section measures of harmonic means of convex bodies , Pacific J. Math., 11 (1961), 1263–1266.
-
[6]
R. J. Gardner, D. Hug, W. Weil , The Orlicz-Brunn-Minkowski theory: A general framework, additions, and inequalities, J. Differential Geom., 97 (2014), 427–476.
-
[7]
C. Haberl, E. Lutwak, D. Yang, G. Zhang, The even Orlicz-Minkowski problem , Adv. Math., 224 (2010), 2485–2510.
-
[8]
M. A. Hernández Cifre, J. Yepes Nicolás , On Brunn-Minkowski type Inequalities for polar bodies, J. Geometric Anal., 26 (2016), 143–155.
-
[9]
A.-J. Li, G. Leng, A new proof of the Orlicz Busemann-Petty centroid inequality, Proc. Amer. Math. Soc., 139 (2011), 1473–1481.
-
[10]
E. Lutwak, Centroid Bodies and Dual Mixed Volumes, Proc. London Math. Soc., 60 (1990), 365–391.
-
[11]
E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom., 38 (1993), 131–150.
-
[12]
E. Lutwak , The Brunn-Minkowski-Firey Theory II: affine and geominimal surface area, Adv. Math., 118 (1996), 244–294.
-
[13]
E. Lutwak, V. Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom., 41 (1995), 227–246.
-
[14]
E. Lutwak, D. Yang, G. Zhang, Lp- affine isoperimetric inequalities, J. Differential Geom., 56 (2000), 111–132.
-
[15]
E. Lutwak, D. Yang, G. Zhang , Orlicz centroid bodies, J. Differential Geom., 84 (2010), 365–387.
-
[16]
E. Lutwak, D. Yang, G. Zhang, Orlicz projection bodies, Adv. Math., 223 (2010), 220–242.
-
[17]
E. Lutwak, G. Zhang , Blaschke Santaló inequalities, J. Differential Geom., 47 (1997), 1–16.
-
[18]
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Second expanded edition, Cambridge University Press, Cambridge (2014)
-
[19]
D. Xi, H. Jin, G. Leng, The Orlicz Brunn-Minkowski inequality, Adv. Math., 260 (2014), 350–374.
-
[20]
G. Y. Zhang, Centered Bodies and Dual Mixed Volumes, Tran. Amer. Math. Soc., 345 (1994), 777–801.
-
[21]
B. Zhu, J. Zhou, W. Xu, Dual Orlicz-Brunn-Minkowski theory , Adv. Math., 264 (2014), 700–725.