Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-18T07:10:05.531Z Has data issue: false hasContentIssue false
  • English
  • Français

BRUNN–MINKOWSKI TYPE INEQUALITIES FOR Lp MOMENT BODIES*

Published online by Cambridge University Press:  25 February 2013

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, lini628@163.com, zhoujz@swu.edu.cn
NI LI
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China e-mail: zhubaocheng814@163.com, lini628@163.com, zhoujz@swu.edu.cn
JIAZU ZHOU
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China e-mail: zhubaocheng814@163.com, lini628@163.com, zhoujz@swu.edu.cn
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.

About 15 years ago, Lutwak and Zhang (E. Lutwak and G. Zhang, Blaschke–Santalo inequalities, J. Differ. Geom. 47 (1997), 1–16) introduced the notion of Lp moment bodies and established important volume inequalities for them, which were recently generalized by Haberl and Schuster (C. Haberl and E. Schuster, General Lp affine isoperimetric inequalities, J. Differ. Geom. 83 (2009), 1–26). In this paper, we establish new Brunn–Minkowski-type inequalities with respect to Blaschke Lp harmonic addition for the quermassintegrals and dual quermassintegrals of Lp moment bodies.

Keywords

52A40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 2 , May 2013 , pp. 391 - 398
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013

Footnotes

*

Research is supported in part by NSFC (10671159 and 11161024).

References

REFERENCES

1.Abardia, J. and Bernig, A., Projection bodies in complex vector spaces, Adv. Math. 227 (2011), 830–846.CrossRefGoogle Scholar
2.Alesker, S., Bernig, A. and Schuster, F. E., Harmonic analysis of translation invariant valuations, Geom. Funct. Anal. 21 (2011), 751–773.CrossRefGoogle Scholar
3.Campi, S. and Gronchi, P., On the reverse Lp Busemann-Petty centroid inequality, Mathematika 49 (2002), 1–11.Google Scholar
4.Campi, S. and Gronchi, P., The Lp Busemann-Petty centriod inequality, Adv. Math. 167 (2002), 128–141.Google Scholar
5.Dafnis, N. and Paouris, G., Small ball probability estimates, ψ2-behavior and the hyperplane conjecture, J. Funct. Anal. 258 (2010), 1933–1964.Google Scholar
6.Firey, W. J., p-means of convex bodies, Math. Scand. 10 (1962), 17–24.Google Scholar
7.Fleury, B., Guedon, O. and Paouris, G., A stability result for mean width of Lp centriod bodies, Adv. Math. 214 (2007), 865–877.Google Scholar
8.Gardner, R. J., Geometric tomography, 2 ed (Cambridge University Press, Cambridge, UK, 2006).Google Scholar
9.Grinberg, E. and Zhang, G., Convolutions, transforms and convex bodies, Proc. Lond. Math. Soc. 78 (1999), 77–115.Google Scholar
10.Haberl, C., Lp intersection bodies, Adv. Math. 217 (2008), 2599–2624.Google Scholar
11.Haberl, C., Star body valued valuations, Indiana Univ. Math. J. 58 (2009), 2253–2276.Google Scholar
12.Haberl, C., Minkowski valuations intertwining the special linear group, J. Eur. Math. Soc. 14 (2012), 1565–1597.Google Scholar
13.Haberl, C. and Schuster, E., General Lp affine isoperimetric inequalities, J. Differ. Geom. 83 (2009), 1–26.CrossRefGoogle Scholar
14.Ludwig, M., Ellipsoids and matrix-valued valuations, Duke Math. J. 119 (2003), 159–188.Google Scholar
15.Ludwig, M., Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), 4191–4213.Google Scholar
16.Lutwak, E., Mixed projection inequalities, Trans. Am. Math. Soc. 287 (1985), 91–106.Google Scholar
17.Lutwak, E., On some affine isoperimetric inequalities, J. Differ. Geom. 23 (1986), 1–13.Google Scholar
18.Lutwak, E., Centroid bodies and dual mixed volumes, Proc. Lond. Math. Soc. 60 (1990), 365–391.Google Scholar
19.Lutwak, E., The Brunn–Minkowski-Firey I: Mixed volumes and the Minkowski problem, J. Differ. Geom. 38 (1993), 131–150.Google Scholar
20.Lutwak, E., The Brunn–Minkowski-Firey II: Affine and geominimal surface areas, Adv. Math. 118 (1996), 244–294.CrossRefGoogle Scholar
21.Lutwak, E., Yang, D. and Zhang, G., Lp affine isoperimetric inequalities, J. Differ. Geom. 56 (2000), 111–132.Google Scholar
22.Lutwak, E., Yang, D. and Zhang, G., Orlicz centroid bodies, J. Differ. Geom. 84 (2010), 365–387.Google Scholar
23.Lutwak, E., Yang, D. and Zhang, G., A volume inequality for polar bodies, J. Differ. Geom. 84 (2010), 163–178.Google Scholar
24.Lutwak, E. and Zhang, G., Blaschke-Santaló inequalities, J. Differ. Geom. 47 (1997), 1–16.Google Scholar
25.Paouris, G., Concentration of mass on isotropic convex bodies, Geom. Funct. Anal. 16 (2006), 1021–1049.CrossRefGoogle Scholar
26.Parapatits, L. and Schuster, F. E., The Steiner formula for Minkowski valuations, Adv. Math. 230 (2012), 978–994.Google Scholar
27.Petty, C. M., Centroid surfaces, Pacific J. Math. 11 (1961), 1535–1547.Google Scholar
28.Petty, C. M., Projection bodies, Proceedings of a colloquium on convexity, Copenhagen (1965) (Kobenhavns Univ. Mat. Inst., Copenhagen, 1967), 234–241.Google Scholar
29.Petty, C. M., Geominimal surface area, Geom. Dedicata. 3 (1974), 77–97.Google Scholar
30.Schneider, R., Convex bodies: The Brunn–Minkowski theory (Cambridge University Press, Cambridge, UK, 1993).Google Scholar
31.Schuster, F. E., Volume inequalities and additive maps of convex bodies, Mathematika 53 (2006), 211–234.Google Scholar
32.Schuster, F. E., Crofton measures and Minkowski valuations, Duke Math. J. 154 (2010), 1–30.Google Scholar
33.Schuster, F. E. and Weberndorfer, M., Volume inequalities for asymmetric Wulff shapes, J. Differ. Geom. arXiv:1110.2069v2.Google Scholar
34.Wang, W. D. and Leng, G. S., Lp dual mixed quermassintegrals, Indian J. Pure Appl. Math. 36 (2005), 177–188.Google Scholar
35.Wannnerer, T., GL(n) equivariant Minkowski valuations, Indiana Univ. Math. J. (to appear).Google Scholar
36.Yaskin, V. and Yaskina, M., Centriod bodies and comparison of volumes, Indiana Univ. Math. J. 55 (2006), 1175–1194.CrossRefGoogle Scholar
37.Yuan, J., Zhao, L. Z. and Leng, G. S., Inequalities for Lp centroid body, Taiwan J. Math. 11 (2007), 1315–1325.Google Scholar
38.Zhao, L. Z., He, B. W. and Leng, G. S., On polars of projection bodies and centroid bodies, Acta Math. Sinica (Chinese Ser.). 49 (2006), 679–686.Google Scholar