Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-21T23:54:27.041Z Has data issue: false hasContentIssue false
  • English
  • Français

ISOPERIMETRIC INEQUALITIES FOR Lp GEOMINIMAL SURFACE AREA*

Published online by Cambridge University Press:  01 August 2011

BAOCHENG ZHU ,
NI LI  and
JIAZU ZHOU
BAOCHENG ZHU
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China e-mail: zhubaocheng814@163.com
NI LI
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
JIAZU ZHOU
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we establish a number of Lp-affine isoperimetric inequalities for Lp-geominimal surface area. In particular, we obtain a Blaschke–Santaló type inequality and a cyclic inequality between different Lp-geominimal surface areas of a convex body.

Keywords

52A40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 717 - 726
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Cianchi, A., Lutwak, E., Yang, D. and Zhang, G., Affine Moser–Trudinger and Morrey–Sobolev inequalities, Calc. Var. Partial Differ. Equ. 36 (2009), 419–436.CrossRefGoogle Scholar
2.Firey, W. J., p-means of convex bodies, Math. Scand. 10 (1962), 17–24.CrossRefGoogle Scholar
3.Gardner, R. J., Geometric tomography (Cambridge University Press, Cambridge, UK, 1995).Google Scholar
4.Haberl, C., Lutwak, E., Yang, D. and Zhang, G., The even Orlicz Minkowski problem, Adv. Math. 224 (2010), 2485–2510.CrossRefGoogle Scholar
5.Haberl, C. and Schuster, F. E., General Lp affine isoperimetric inequalities, J. Differ. Geom. 83 (2009), 1–26.CrossRefGoogle Scholar
6.Haberl, C. and Schuster, F. E., Asymmetric affine Lp Sobolev inequalities, J. Funct. Anal. 257 (2009), 641–658.CrossRefGoogle Scholar
7.Haberl, C., Schuster, F. and Xiao, J., An asymmetric affine Pólya–Szegö principle, arXiv: 0908.1557.Google Scholar
8.Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge University Press, London, 1934).Google Scholar
9.Leichtweiß, K., On the history of the affine surface area for convex bodies, Results Math. 20 (1991), 650–656.CrossRefGoogle Scholar
10.Leng, G., Affine surface areas of curvature images for convex bodies, Acta. Math. Sin. 45 (4) (2002), 797–802 (in Chinese).Google Scholar
11.Ludwig, M., General affine surface areas, Adv. Math. 224 (2010), 2346–2360.CrossRefGoogle Scholar
12.Ludwig, M. and Reitzner, M., A characterization of affine surface area, Adv. Math. 147 (1999), 138–172.CrossRefGoogle Scholar
13.Ludwig, M. and Reitzner, M., A classification of SL(n) invariant valuations, Ann. Math. 172 (2010), 1223–1271.CrossRefGoogle Scholar
14.Lutwak, E., Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232–261.CrossRefGoogle Scholar
15.Lutwak, E., Extended affine surface area, Adv. Math. 85 (1991), 39–68.CrossRefGoogle Scholar
16.Lutwak, E., The Brunn–Minkowski–Firey theory I: Mixed volumes and the Minkowski problem, J. Differ. Geom. 38 (1993), 131–150.CrossRefGoogle Scholar
17.Lutwak, E., The Brunn–Minkowski-Firey theory II: Affine and geominimal surface areas, Adv. Math. 118 (1996), 244–294.CrossRefGoogle Scholar
18.Lutwak, E., Yang, D. and Zhang, G. Y., Lp affine isoperimetric inequalities, J. Differ. Geom. 56 (2000), 111–132.CrossRefGoogle Scholar
19.Lutwak, E., Yang, D. and Zhang, G., Sharp affine Lp Sobolev inequalities, J. Differ. Geom. 62 (2002), 17–38.CrossRefGoogle Scholar
20.Lutwak, E., Yang, D. and Zhang, G., Orlicz projection bodies, Adv. Math., 223 (2010), 220–242.CrossRefGoogle Scholar
21.Lutwak, E., Yang, D. and Zhang, G., Orlicz centroid bodies, J. Differ. Geom. 84 (2010), 365–387.CrossRefGoogle Scholar
22.Lutwak, E. and Zhang, G. Y., Blaschke–Santaló inequalities, J. Differ. Geom. 47 (1997), 1–16.CrossRefGoogle Scholar
23.Lv, S. and Leng, G., The Lp-curvature images of convex bodies and Lp-projection bodies, Proc. Indian Acad. Sci. Math. Sci. 118 (2008), 413–424.CrossRefGoogle Scholar
24.Petty, C. M., Affine isoperimetric problems, Ann. N. Y. Acad. Sci. 440 (1985), 113–127.CrossRefGoogle Scholar
25.Petty, C. M., Isoperimetric problems, in Proceedings of the Conference on Convexty and Combinatorial Geometry (University of Oklahoma, Norman, OK, 1971, 1972), pp. 26–41.Google Scholar
26.Petty, C. M., Geominimal surface area, Geom. Dedicata 3 (1974), 77–97.CrossRefGoogle Scholar
27.Schneider, R., Convex bodies: the Brunn–Minkowski theory, (Cambridge University Press, Cambridge, UK, 1993.CrossRefGoogle Scholar
28.Schuster, F. E., Volume inequalities and additive maps of convex bodies, Mathematika 53 (2006), 211–234.CrossRefGoogle Scholar
29.Schuster, F. E., Convolutions and multiplier transformations of convex bodies, Trans. Amer. Math. Soc. 359 (2007), 5567–5591.CrossRefGoogle Scholar
30.Schuster, F. E. and Wannerer, T., GL(n) contravariant Minkowski valuations, Trans. Amer. Math. Soc., to appear.Google Scholar
31.Stancu, A. and Werner, E., New higher-order equiaffine invariants, Isr. J. Math. 171 (2009), 221–235.CrossRefGoogle Scholar
32.Wang, W. and Leng, G., Some affine isoperimetric inequalities associated with Lp-affine surface area, Houston J. Math. 34 (2008), 443–453.Google Scholar
33.Werner, E., On Lp-affine surface areas, Indiana Univ. Math. J. 56 (2007), 2305–2323.CrossRefGoogle Scholar
34.Werner, E. and Ye, D., New Lp affine isoperimetric inequalities, Adv. Math. 218 (2008), 762–780.CrossRefGoogle Scholar
35.Yuan, J., Lv, L. and Leng, G., The p-affine surface area, Math. Ineq. Appl. 3 (2007), 693–702.Google Scholar