EXTREMUM PROPERTIES OF DUAL Lp-CENTROID BODY AND Lp-JOHN ELLIPSOID

Title & Authors
EXTREMUM PROPERTIES OF DUAL Lp-CENTROID BODY AND Lp-JOHN ELLIPSOID
Ma, Tong-Yi;

Abstract
For $\small{0}$<$\small{p{\leq}{\infty}}$ and a convex body $\small{K}$ in $\small{\mathbb{R}^n}$, Lutwak, Yang and Zhang defined the concept of dual $\small{L_p}$-centroid body $\small{{\Gamma}_{-p}K}$ and $\small{L_p}$-John ellipsoid $\small{E_pK}$. In this paper, we prove the following two results: (i) For any origin-symmetric convex body $\small{K}$, there exist an ellipsoid $\small{E}$ and a parallelotope $\small{P}$ such that for $\small{1{\leq}p{\leq}2}$ and $\small{0}$<$\small{q{\leq}{\infty}}$, $\small{E_qE{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qP}$ and $\small{V(E)=V(K)=V(P)}$; For $\small{2{\leq}p{\leq}{\infty}}$ and $\small{0}$<$\small{q{\leq}{\infty}}$, $\small{2^{-1}{\omega_n}^{\frac{1}{n}}E_qE{\subseteq}{\Gamma}_{-p}K{\subseteq}{2\omega_n}^{-\frac{1}{n}}(nc_{n-2,p})^{-\frac{1}{p}}E_qP}$ and $\small{V(E)=V(K)=V(P)}$. (ii) For any convex body $\small{K}$ whose John point is at the origin, there exists a simplex $\small{T}$ such that for $\small{1{\leq}p{\leq}{\infty}}$ and $\small{0}$<$\small{q{\leq}{\infty}}$, $\small{{\alpha}n(nc_{n-2,p})^{-\frac{1}{p}}E_qT{\supseteq}{\Gamma}_{-p}K{\supseteq}(nc_{n-2,p})^{-\frac{1}{p}}E_qT}$ and $\small{V(K)=V(T)}$.
Keywords
John ellipsoid;$\small{L_p}$-John ellipsoid;new ellipsoid;dual $\small{L_p}$-centroid body;simplex;
Language
English
Cited by
References
1.
K. M. Ball, An elementary introduction to modern convex geometry, Flavors of geometry, 1-58, Math. Sci. Res. Inst. Publ., 31, Cambridge Univ. Press, Cambridge, 1997.

2.
J. Jonasson, Optimization of shape in continuum percolation, Ann. Probab. 29 (2001), no. 2, 624-635.

3.
F. John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, 187-204. Inter- science Publishers, Inc., New York, N. Y., 1948.

4.
K. Leichtweifi, Affine Geometry of Convex Bodies, J. A. Barth, Heidelberg, 1998.

5.
J. Lindenstrauss and V. D. Milman, Local Theory of Normal Spaces and Convexity, Handbook of Convex Geometry (P. M. Gruber and J. M. Wills, eds.), North-Holland, Amsterdam, 1993, 1149-1220.

6.
E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131-150.

7.
E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. in Math. 118 (1996), no. 2, 244-294.

8.
E. Lutwak, D. Yany, and G. Y. Zhang, A New Ellipsoid Associated with Convex Bodies, Duke Math. J. 104 (2000), no. 3, 375-390.

9.
E. Lutwak, D. Yany, and G. Y. Zhang, Lp John ellipsoids, Proc. London Math. Soc. (3) 90 (2005), no. 2, 497-520.

10.
E. Lutwak, D. Yany, and G. Y. Zhang, $L_{p}$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111-132.

11.
E. Lutwak and G. Y. Zhang, Blaschke-Santalo inequalities, J Differential Geom. 47 (1997), no. 1, 1-16.

12.

13.
V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, Geometric aspects of functional analysis (1987-1988), 64-104, Lecture Notes in Math., 1376, Springer, Berlin, 1989.

14.
R. Schneider, Simplices, Program \Asymptotic Theory of the Geometry of Finite Dimensional Spaces", Schrodinger Institute, Vienna, July 2005.

15.
W. D. Wang, Extremum Problems in Convex Bodies Geometry of $L_{p}$-space, A Dissertation Submitted To Shanghai University for the Degree of Doctor in Science, February, 2006.

16.
W. D. Wang and G. S. Leng, On Some Inequalities for the New Geometric Body ${\Gamma}-_{p}K$. Acta Mathematica Sinica, Chinese Series 49 (2006), no. 6, 1327-1334.

17.
J. Yuan, L. Si, and G. S. Leng, Extremum properties of the new ellipsoid, Tamkang J. Math. 38 (2007), no. 2, 159-165.