The mixed \(L_p\)-dual affine surface area for multiple star bodies

Volume 9, Issue 5, pp 2813--2822 http://dx.doi.org/10.22436/jnsa.009.05.76

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1508 Downloads
• 2380 Views

Authors

Zhang Ting - Department of Mathematics, China Three Gorges University, Yichang, 443002, P. R. China. Wang Weidong - Department of Mathematics, China Three Gorges University, Yichang, 443002, P. R. China. Si Lin - Department of Mathematics, Beijing Forestry University, Beijing, 100083, P. R. China.

Abstract

Associated with the notion of the mixed \(L_p\)-affine surface area for multiple convex bodies for all real \(p (p \neq -n)\) which was introduced by Ye, et al. [D. Ye, B. Zhu, J. Zhou, arXiv, 2013 (2013), 38 pages], we define the concept of the mixed \(L_p\)-dual affine surface area for multiple star bodies for all real \(p (p \neq -n)\) and establish its monotonicity inequalities and cyclic inequalities. Besides, the Brunn-Minkowski type inequalities of the mixed \(L_p\)-dual affine surface area for multiple star bodies with two addition are also presented.

Share and Cite

Keywords

• \(L_p\)-affine surface area
• \(L_p\)-dual affine surface area
• multiple star bodies
• Hölder inequality.

MSC

•  52A20
•  52A40.

References

• [1] W. J. Firey, Polar means of convex bodies and a dual to the Brunn-Minkowski theorem , Canad. J. Math., 13 (1961), 444-453.


• [2] W. J. Firey, Mean cross-section measures of harmonic means of convex bodies, Pacific J. Math., 11 (1961), 1263-1266.


• [3] W. J. Firey, p-Means of convex bodies , Math. Scandinavica, 10 (1962), 17-24.


• [4] R. J. Gardner , Geometric tomography, Cambrige University Press, Cambrige (1995)


• [5] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambrige University Press, Cambrige (1952)


• [6] J. C. Kuang, Applied inequalities, Shangdong Science and Technology Press, Jinan (2010)


• [7] K. Leichtweiss, Bemerkungen zur Definition einer erweiterten Affinoberäche von E.Lutwak, Manuscripta Math., 65 (1989), 181-197.


• [8] K. Leichtweiss, On the history of the affine surface area for convex bodies, Results Math., 20 (1991), 650-656.


• [9] E. Lutwak, On the Blaschke-Santaló inequality, Anal. New York Acad. Sci., 440 (1985), 106-112.


• [10] E. Lutwak, On some affine isoperimetric inequalities, J. Differ. Geom, 23 (1986), 1-13.


• [11] E. Lutwak, Mixed affine surface area, J. Math. Anal. Appl., 125 (1987), 351-360.


• [12] E. Lutwak, Extended affine surface area, Adv. Math., 85 (1991), 39-68.


• [13] E. Lutwak, The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem, J. Differ. Geom, 38 (1993), 131-150.


• [14] E. Lutwak, The Brunn-Minkowski-Firey theory II: affine and geominimal surface areas, Adv. Math., 118 (1996), 244-294.


• [15] R. Schneider, Convex Bodies: The Brunn-Minkowski theory, Cambridge University Press, Cambridge (2014)


• [16] C. Schütt, E. Werner , Surface bodies and p-affine surface area , Adv. Math., 187 (2004), 98-145.


• [17] W. Wang, G. Leng, \(L_p\)-mixed affine surface area, J. Math. Anal. Appl., 335 (2007), 341-354.


• [18] J. Y. Wang, W. D. Wang, L-dual affine surface area forms of Busemann-Petty type problems, Proc. Indian Acad. Sci., 125 (2015), 71-77.


• [19] W. Wei, H. Binwu , \(L_p\)-dual affine surface area, J. Math. Anal. Appl., 348 (2008), 746-751.


• [20] W. Wei, Y. Jun, H. Binwu, Inequalities for \(L_p\)-dual affine surface area, Math. Inequal. Appl., 13 (2010), 319-327.


• [21] E. M. Werner, On \(L_p\) affine surface areas , Indiana Univ. Math. J., 56 (2007), 2305-2324.


• [22] E. M. Werner, Rényi divergence and \(L_p\)-affine surface area for convex bodies, Adv. Math., 230 (2012), 1040-1059.


• [23] E. Werner, D. Ye, New \(L_p\)-affine isoperimetric inequalities, Adv. Math., 218 (2008), 762-780.


• [24] E. Werner, D. Ye , Inequalities for mixed p-affine surface area , Math. Ann., 347 (2010), 703-737.


• [25] D. Ye , Inequalities for general mixed affine surface areas, J. London Math. Soc., 85 (2012), 101-120.


• [26] D. Ye, B. Zhu, J. Zhou, The mixed \(L_p\) geominimal surface areas for multiple convex bodies, arXiv, 2013 (2013), 38 pages.