Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T03:57:12.581Z Has data issue: false hasContentIssue false
  • English
  • Français

AN EXTENDED LOOMIS–WHITNEY INEQUALITY FOR POSITIVE DOUBLE JOHN BASES

Published online by Cambridge University Press:  10 March 2011

AI-JUN LI ,
GUANGTING WANG  and
GANGSONG LENG
AI-JUN LI
Affiliation:
School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo City 454000, China e-mail: liaijun72@163.com
GUANGTING WANG
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, China e-mail: tinggw@163.com
GANGSONG LENG
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, China e-mail: gleng@staff.shu.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.

In this paper, we establish an extended Loomis–Whitney inequality for positive double John bases, which generalises Ball's result [1]. Moreover, a different extension of the Loomis–Whitney inequality is deduced.

Keywords

52A2052A2152A40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 451 - 462
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Ball, K., Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), 891–901.CrossRefGoogle Scholar
2.Ball, K., Volume ratios and a reverse isoperimetric inequality, J. Lond. Math. Soc. 44 (1991), 351–359.CrossRefGoogle Scholar
3.Ball, K., An elementary introduction to modern convex geometry, Flavors of geometry, in Mathematical Science Research Institute Publication, Vol. 31 (Levy, S., Editor) (Cambridge University Press, Cambridge, UK, 1997), 1–58.Google Scholar
4.Bastero, J. and Romance, M., John's decomposition of the identity in the non-convex case, Positivity 6 (2002), 1–161.CrossRefGoogle Scholar
5.Burago, Yu. D. and Zalgaller, V. A., Geometric inequalities (Springer, Berlin, 1988).CrossRefGoogle Scholar
6.Gardner, R. J., Geometric tomography (Cambridge University Press, Cambridge, UK, 1995).Google Scholar
7.Giannopoulos, A., Perissinaki, I. and Tsolomitis, A., John's theorem for an arbitrary pair of convex bodies, Geom. Dedicata 84 (2001), 63–79.CrossRefGoogle Scholar
8.Gordon, Y., Litvak, A. E., Meyer, M. and Pajor, A., John's decomposition in the general case and applications, J. Differ. Geom. 68 (1) (2004), 99–119.CrossRefGoogle Scholar
9.Gruber, P. M. and Schuster, F., An arthmetic proof of John's ellipsoid theorem, Arch. Math. 85 (2005), 82–88.CrossRefGoogle Scholar
10.Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge University Press, Cambridge, UK, 1959).Google Scholar
11.John, F., Extremum problems with inequalities as subsidiary conditions, in Studies and essays presented to R. Courant on his 60th birthday, January 8, 1948 (Interscience Publishers, New York, NY, 1948), 187–204.Google Scholar
12.Lewis, D., Ellipsoids defined by Banach ideal norms, Mathematika 26 (1979), 18–29.CrossRefGoogle Scholar
13.Loomis, L. H. and Whitney, H., An inequality related to the isoperimetric inequality, Bull. Amer. Math. Soc. 55 (1949), 961–962.CrossRefGoogle Scholar
14.Milman, V. D. and Pajor, A., Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in Geometric aspects of functional analysis (Lindenstrauss-Milman, , Editor), Lecture Notes in Math., 1376 (Springer, Berlin, 1989), 64–104.CrossRefGoogle Scholar
15.Pisier, G., The volume of convex bodies and Banach space geometry, in Cambridge tracts in mathematics, Vol. 94, (Cambridge University Press, Cambridge, UK, 1989).Google Scholar
16.Schneider, R., Convex bodies: The Brunn–Minkowski theory, in Encyclopedia of mathematics and its applications, Vol. 44, (Cambridge University Press, Cambridge, UK, 1993).Google Scholar
17.Tomczak-Jaegermann, N., Banach–Mazur distances and finite-dimensional operator ideals, in Pitman monographs and surveys in pure and applied mathematics, Vol. 38 (Pitman, London, 1989).Google Scholar
18.Zhang, G., The affine Sobolev inequality, J. Differ. Geom. 53 (1999), 183–202.CrossRefGoogle Scholar