Abstract
In this paper, Brunn-Minkowski inequality and Dresher-type inequality for mixed width-integrals of Firey’s p-sum are established. Further, we present the Dresher-type inequalities for dual quermassintegrals of the polar of Lp projection body and Lp centroid body, which in special cases yield some previous inequalities.
Similar content being viewed by others
References
Blaschke W. Vorlesungen über lntegralgeometrie [M]. New York: Springer-Verlag, 1949.
Hadwiger H. Vorlesungen über lnhalt, Oberfl ache undIsoperimetrie [M]. Berlin: Springer-Verlag, 1957.
Chakerian G D. The mean volume of boxes and cylinders circumscribed about a convex body [J]. Israel J Math, 1972, 12: 249–256.
Chakerian G D. Isoperimetric inequalities for the mean width of a convex body [J]. Geom Dedicata, 1973, 1: 356–362.
Lutwak E. Width-integrals of convex bodies [J]. Proc Amer Math Soc, 1975, 53: 435–439.
Lutwak E. Mixed width-integrals of convex bodies [J]. Israel J Math, 1977, 28(3): 249–253.
Cheung W S, Zhao C J. Width-integrals and affine surface area of convex bodies [J]. Banach J Math Anal, 2008, 316: 70–77.
Leng G S, Zhao C J, He B W, et al. Inequalities for polars of mixed projection bodies [J]. Sci China Ser A, 2004, 47(2): 175–186.
Lutwak E. Mixed projection inequalities [J]. Trans Amer Math Soc, 1985, 287: 91–106.
Lv S J. Dual Brunn-Minkowski inequality for volume differences [J]. Geom Dedicata, 2010, 145: 169–180.
Zhao C J. The Brunn-Minkowski type inequality [J]. Bull Iran Math Soc, 2010, 36: 157–167.
Zhao C J, Leng G S. On polars of mixed projection bodies [J]. J Math Anal Appl, 2006, 316: 664–678.
Li N, Zhu B C. Mixed brightness integrals of convex bodies [J]. J Korean Math Soc, 2010, 47(5): 935–945.
Lutwak E. The Brunn-Minkowski-Firey theory I: Mixed volumes and the minkowski problem [J]. J Differential Geom, 1993, 38: 131–150.
Lutwak E. The Brunn-Minkowski-Firey theory II: Affine and geominimal surface areas [J]. Adv Math, 1996, 118: 244–294.
Campi S, Grochi P. The Lp-Busemann-Petty centroid inequality [J]. Adv Math, 2002, 167(1): 128–141.
Campi S, Grochi P. On the reverse Lp-Busemann-Petty centroid inequality [J]. Mathematika, 2002, 49: 1–11.
Haberl C, Schuster F. General Lp affine isoperimetric inequalities [J]. J Differential Geom, 2009, 83: 1–26.
Lutwak E, Yang D, Zhang G Y. Lp affine isoperimetric inequalities [J]. J Differential Geom, 2000, 56: 111–132.
Lutwak E, Zhang G Y. Blaschke-Santal ó inequalities [J]. J Differential Geom, 1997, 47: 1–16.
Wang W D, Leng G S. On the Lp-versions of the Petty’s conjectured projection inequality and applications [J]. Taiwan J Math, 2008, 12: 1067–1086.
Wang W D, Wei D J, Xiang Y. Some inequalities for the Lp-curvature image [J]. J Inequal Appl, 2009, 6: 1–10.
Yuan J, Zhao L Z, Leng G S. Inequalities for Lp centroid body [J]. Taiwan J Math, 2007, 11: 1315–1325.
Gardner R J. Geometric Tomography [M]. 2nd Ed. Cambridge: Cambridge Univ Press, 2006.
Schneider R. Convex Bodies: The Brunn Minkowski Theory [M]. 2nd Ed. Cambridge: Cambridge University Press, 2014.
Lutwak E. Dual mixed volumes [J]. Pacific J Math, 1975, 58: 531–538.
Dresher M. Moment space and inequalities [J]. Duke Math J, 1953, 20: 261–271.
Zhao C J. On Dresher’s inequalities for width-integrals [J]. Appl Math Lett, 2012, 25: 190–194.
Author information
Authors and Affiliations
Corresponding author
Additional information
Foundation item: Supported by the National Natural Science Foundation of China (11371224), the Innovation Foundation of Graduate Student of China Three Gorges University (2014CX097) and Excellent Foundation of Degree Dissertation of Master of China Three Gorges University (2015PY071)
Biography: ZHANG Ting, female, Master candidate, research direction: convex geometric analysis.
Rights and permissions
About this article
Cite this article
Zhang, T., Wang, W. Inequalities for mixed width-integrals. Wuhan Univ. J. Nat. Sci. 21, 185–190 (2016). https://doi.org/10.1007/s11859-016-1157-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11859-016-1157-6
Key words
- mixed width-integrals
- Lp projection body
- Lp centroid body
- Brunn-Minkowski inequality
- Dresher-type inequality