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Lp-DUAL MIXED GEOMINIMAL SURFACE AREA

Published online by Cambridge University Press:  02 September 2013

YIBIN FENG  and
WEIDONG WANG
YIBIN FENG
Affiliation:
Department of Mathematics, China Three Gorges University, Yichang 443002, China e-mails: [email protected]; [email protected]
WEIDONG WANG
Affiliation:
Department of Mathematics, China Three Gorges University, Yichang 443002, China e-mails: [email protected]; [email protected]
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Abstract

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Lutwak (Adv. Math., vol. 118(2), 1996, pp. 244–294) defined the notion of Lp-geominimal surface area based on Lp-mixed volumes. Recently, Wang and Qi (J. Inequal. Appl., vol. 2011, 2011, pp. 1–10) introduced the concept of Lp-dual geominimal surface area based on Lp-dual mixed volumes. In this paper, based on Lp-dual mixed quermassintegrals, we define the concept of Lp-dual mixed geominimal surface area and establish several inequalities for this new notion.

Keywords

52A2052A40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 56 , Issue 1 , January 2014 , pp. 229 - 239
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Alesker, S., Bernig, A. and Schuster, F. E., Harmonic analysis of translation invariant valuations, Geom. Funct. Anal. 21 (2011), 751–773.Google Scholar
2.Gardner, R. J., Geometric tomography, 2nd edn. (Cambridge University Press, Cambridge, UK, 2006).Google Scholar
3.Gardner, R. J., Koldobsky, A. and Schlumprecht, T., An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. Math. 149 (1999), 691–703.Google Scholar
4.Haberl, C., L p-intersection bodies, Adv. Math. 217 (6) (2008), 2599–2624.CrossRefGoogle Scholar
5.Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge University Press, Cambridge, UK, 1959).Google Scholar
6.Ludwig, M., Intersection bodies and valuations, Amer. J. Math. 128 (2006), 1409–1428.CrossRefGoogle Scholar
7.Ludwig, M., General affine surface areas, Adv. Math. 224 (2010), 2346–2360.Google Scholar
8.Ludwig, M. and Reitzner, M., A characterization of affine surface area, Adv. Math. 147 (1999), 138–172.Google Scholar
9.Ludwig, M. and Reitzner, M., A classification of SL(n) invariant valuations, Ann. Math. 172 (2010), 1223–1271.Google Scholar
10.Lutwak, E., Dual mixed volumes, Pacific J. Math. 58 (1975), 531–538.CrossRefGoogle Scholar
11.Lutwak, E., Volume of mixed bodies, Trans. Amer. Math. Soc. 294 (2) (1986), 487–500.CrossRefGoogle Scholar
12.Lutwak, E., Mixed affine surface area, J. Math. Anal. Appl. 125 (1987), 351–360.CrossRefGoogle Scholar
13.Lutwak, E., Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232–261.Google Scholar
14.Lutwak, E., Centroid bodies and dual mixed volumes, Proc. Lond. Math. Soc. 60 (1990), 365–391.CrossRefGoogle Scholar
15.Lutwak, E., The Brunn–Minkowski–Firey theory II: Affine and geominimal surface areas, Adv. Math. 118 (2) (1996), 244–294.CrossRefGoogle Scholar
16.Parapatits, L. and Schuster, F. E., The Steiner formula for Minkowski valuations, Adv. Math. 230 (2012), 978–994.Google Scholar
17.Petty, C. M., Geominimal surface area, Geom. Dedicata 3 (1) (1974), 77–97.CrossRefGoogle Scholar
18.Schneider, R., Convex bodies: the Brunn–Minkowski theory (Cambridge University Press, Cambridge, UK, 1993).Google Scholar
19.Schuster, F. E., Volume inequalities and additive maps of convex bodies, Mathematika 53 (2006), 211–234.Google Scholar
20.Schuster, F. E., Crofton measures and Minkowski valuations, Duke Math. J. 154 (2010), 1–30.Google Scholar
21.Stancu, A. and Werner, E., New higher-order equiaffine invariants, Israel J. Math. 171 (2009), 221–235.Google Scholar
22.Wang, W. D. and Feng, Y. B., A general L p-version of Petty's affine projection inequality, Taiwan J. Math. 17 (2) (2013), 517–528.CrossRefGoogle Scholar
23.Wang, W. D. and Leng, G. S., L p-dual mixed quermassintegrals, Indian J. Pure Appl. Math. 36 (4) (2005), 177–188.Google Scholar
24.Wang, W. D. and Qi, C., L p-dual geominimal surface area, J. Inequal. Appl. 2011 (2011), 1–10.Google Scholar
25.Werner, E., On L p-affine surface areas, Indiana Univ. Math. J. 56 (2007), 2305–2323.Google Scholar
26.Werner, E. and Ye, D., New L p-affine isoperimetric inequalities, Adv. Math. 218 (2008), 762–780.Google Scholar
27.Zhang, G. Y., A positive solution to the Busemann–Petty problem in ℝ4, Ann. Math. 149 (1999), 535–543.Google Scholar
28.Zhu, B. C., Li, N. and Zhou, J. Z., Isoperimetric inequalities for L p-geominimal surface area, Glasg. Math. J. 53 (2011), 717–726.Google Scholar