凸体的对偶迷向常数与LYZ椭球

熊革, 胡家麒

数学学报 ›› 2014, Vol. 57 ›› Issue (5) : 947-960.

PDF(570 KB)
PDF(570 KB)
数学学报 ›› 2014, Vol. 57 ›› Issue (5) : 947-960. DOI: 10.12386/A2014sxxb0088
论文

凸体的对偶迷向常数与LYZ椭球

    熊革, 胡家麒
作者信息 +

The Dual Isotropic Constant and the LYZ Ellipsoid of Convex Bodie

    Ge XIONG, Jia Qi HU
Author information +
文章历史 +

摘要

研究了凸体处于对偶迷向位置时的解析特征,并建立了凸体对偶迷向常数的新的下界;其次,证明了关于原点中心对称凸体的LYZ椭球与John椭球相等的充要条件;最后,举例 具体计算了几个凸多边形的 LYZ椭球和John椭球,以进一步认清两者的差别.

Abstract

We mainly study the analytical characterizations of convex bodies when they are positioned in the dual isotropic position. Several new lower bounds are established for the dual isotropic constant. Also we find the necessary and sufficient conditions such that the LYZ ellipsoid and the John ellipsoid are the same. To further distinguish the two important ellipsoids, we calculate and depict the LYZ ellipsoids and John ellipsoids of several specific convex polygons.

关键词

LYZ 椭球 / John 椭球 / Lp John 椭球 / 对偶迷向位置 / 对偶迷向常数

Key words

LYZ ellipsoid / John ellipsoid / Lp John ellipsoid / dual isotropic position / dual isotropic constant

引用本文

导出引用
熊革, 胡家麒. 凸体的对偶迷向常数与LYZ椭球. 数学学报, 2014, 57(5): 947-960 https://doi.org/10.12386/A2014sxxb0088
Ge XIONG, Jia Qi HU. The Dual Isotropic Constant and the LYZ Ellipsoid of Convex Bodie. Acta Mathematica Sinica, Chinese Series, 2014, 57(5): 947-960 https://doi.org/10.12386/A2014sxxb0088

参考文献

[1] Ball K. M., An Elementary Introduction to Modern Convex Geometry, Flavors of Geometry, CambridgeUniversity Press, New York, 1997.

[2] Ball K. M., Ellipsoids of maximal volume in convex bodies, Geom. Dedicata., 1992, 41: 241-250.

[3] Ball K. M., Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc., 1991, 44(2): 351-359.

[4] Barthe F., On a reverse form of the Brascamp-Lieb inequality, Invent. Math., 1998, 134: 335-361.

[5] Gardner R. J., Geometric Tomography, Gambridge University Press, Gambridge, UK, 1995.

[6] John F., Extremum Problems with Inequalities as Subsidiary Conditions, Courant Anniversary Volume.,Interscience, New York, 1948, 187-204.

[7] Lutwak E., The Brunn-Minkowski-Firey theory, I, Mixed volumes and the Minkowski problem, J. DifferentialGeom., 1993, 38: 131-150.

[8] Lutwak E., The Brunn-Minkowski-Firey theory, II, Affine and geominimal surface areas, Adv. Math., 1996,118: 244-294.

[9] Lutwak E., Yang D., Zhang G. Y., A new ellipsoid associated with convex bodies, Duke Math. J., 2000, 104:375-390.

[10] Lutwak E., Yang D., Zhang G. Y., A volume inequality for polar bodies, J. Differential Geom., 2010, 84:163-178.

[11] Lutwak E., Yang D., Zhang G. Y., Lp John ellipsoids, Proc. Lond. Math. Soc., 2005, 90: 497-520.

[12] Lutwak E., Yang D., Zhang G. Y., Volume inequalities for isotropic measures, Amer. J. Math., 2007, 129:1711-1723.

[13] Lutwak E., Yang D., Zhang G. Y., Volume inequalities for subspaces of Lp, J. Differential Geom., 2004, 68:159-184.

[14] Milman V. D., Pajor A., Isotropic Position and Inertia Ellipsoids and Zonoids of the unit Ball of a Normedn-Dimensional Space, Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1376, Springer,Berlin, 1989: 64-104.

[15] Schneider R., Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press, Cambridge, UK,1993.

基金

国家自然科学基金资助项目(11001163);上海市教委科研创新基金资助项目(11YZ11)

PDF(570 KB)

Accesses

Citation

Detail

段落导航
相关文章

/