关于Lp -截面体和Lp -混合截面体

数学物理学报 ›› 2012, Vol. 32 ›› Issue (4): 753-767.

关于L_p -截面体和L_p -混合截面体

马统一

河西学院数学与统计学院甘肃张掖 734000

收稿日期:2011-05-29 修回日期:2012-04-27 出版日期:2012-08-25 发布日期:2012-08-25
基金资助:
国家自然科学基金(11161019)和甘肃省教育厅研究生导师科研基金(1009B-09)资助

On L_p-Intersection Body and L_p-Mixed Intersection Body

MA Tong-Yi

College of Mathematics and Statistics, Hexi University, Gansu Zhangye 734000

Received:2011-05-29 Revised:2012-04-27 Online:2012-08-25 Published:2012-08-25
Supported by:
国家自然科学基金(11161019)和甘肃省教育厅研究生导师科研基金(1009B-09)资助

摘要/Abstract

摘要：

对于p<1和p≠0, Haberl和Ludwig引进了星体的L_p -截面体I_pK的概念. 该文研究截面体的极值性质, 获得了L_p -截面体的单调性, 建立了L_p-Busemann-Petty截面不等式.并且将L_p -截面体的概念进一步拓展, 提出了L_p -混合截面体的概念. 作为应用, 建立了L_p -混合截面体和它的极体的Aleksandrov -Fenchel型不等式. 这些结果是已有结果的对偶形式.

关键词: 面体, L_p -截面体, L_p -混合截面体

Abstract:

For p<1and p≠0, Haberl and Ludwig introduced an L_p-intersection body I_pK of a star body K. In this paper we study extreme nature of the
L_p-intersection body, and L_p-type Busemann-Petty intersection inequality is established. Meanwhile, we further expand the concept of
L_p-intersection body to L_p-mixed intersection body is put forward. As an application, we establish Aleksandrov-Fenchel type inequalities for L_p-mixed intersection body and its polar body. These results are the dual form of some known results.

Key words: Intersection body, L_p-intersection body, L_p-mixed intersection body

中图分类号:

52A40

马统一. 关于L_p -截面体和L_p -混合截面体[J]. 数学物理学报, 2012, 32(4): 753-767.

MA Tong-Yi. On L_p-Intersection Body and L_p-Mixed Intersection Body[J]. Acta mathematica scientia,Series A, 2012, 32(4): 753-767.

参考文献

[1] Lutwak E. Intersection bodies and dual mixed volumes. Adv Math, 1988, 71: 232--261

[2] Goodey P, Lutwak E, Weil W. Functional analytic characterizations of classes of convex bodies. Math Z, 1996, 222: 363--381

[3] Zhang G. A positive solution to the Busemann-Petty problem in R⁴. Ann Math, 1999, 149: 535--543

[4] Gardner R J. Apositive answer to the Busemann-Petty problem in three dimensions. Ann Math, 1994, 140: 435--447

[5] Gardner R J, Koldobsky A, Schlumroecht T. An analytic solution to the Busemann-Petty problem on sections of convex bodies.
Ann Math, 1999, 149: 691--703

[6] Busemann H.Volume in terms of concurrent cross-sections. Pacific J Math, 1953, 3: 1--12

[7] Campi S. Stability estimates for star bodies in terms of their intersection bodies. Mathematika, 1998, 45: 287--303

[8] Campi S. Convex intersection bodies in three and four dimensions. Mathematika, 1999, 46: 15--27

[9] Fallert H, Goodey P, Weil W. Spherical projections and centrally symmetric sets. Adv Math, 1997, 129: 301--322

[10] Gardner R J. A positive answer to the Busemann-Petty problem in three dimensions. Ann Math, 1994, 140: 435--447

[11] Gardner R J. On the Busemann-petty problem concerning central sections of centrally symmetric convex bodies. Bull Amer Math Soc (NS), 1994, 30: 222--226

[12] Gardner R J. Intersection bodies and the Busemann-petty problem. Trans Amer Math Soc, 1994, 342: 435--445

[13] Gardner R J. Geometric Tomography. Cambridge: Cambridge University Press, 1995

[14] Gardner R J, Koldobsky A, Schlumprecht T. An analytic solution to the Busemann-petty problem. C R Acad Sci Paris S I Math, 1999, 328: 29--34

[15] Goodey P,Weil W. Intersection bodies and ellipsoids. Mathematika, 1995, 42: 295--304

[16] Grinberg E, Zhang G. Convolutions, transforms, and convex bodies. Proc London Math soc, 1999, 78: 77--115

[17] Koldobsky A. A functional analytic approach to intersection bodies. Geom Funct Anal, 2000, 10: 1507--1526

[18] Koldobsky A. Intersection bodies, positive definite distributions, and the Busemann-Petty problem. Amer J Math, 1998, 120: 827--840

[19] Koleobsky A. Second derivative test for intersection bodies. Adv Math, 1998, 136: 15--25

[20] Koldobsky A. Intersection bodies in R⁴. Adv Math, 1998, 136: 1--14

[21] Koldobsky A. Intersection bodies and the Busemann-petty problem. C R Acad Sci Paris S I Math, 1997, 325: 1181--1186

[22] Ludwig M. Intersection bodies and valuations. Amer J Math, 2006, 128: 1409--1428

[23] Moszynska M. Quotient star bodies, intersection bodies, and star duality. J Math Anal Appl, 1999, 232: 45--60

[24] Zhang G. Iintersection bodies and polytopes. Mathematika, 1999, 46: 29--34

[25] Zhang G. Sections of convex bodies. Amer J Math, 1996, 118: 319--340

[26] Zhang G. Intersection bodies and the Busemann-Petty inequalitiez in R⁴. Ann Math, 1994, 140: 331--346

[27] Zhang G. Centered bodies and dual mixed volumes. Trans Amer Math Soc, 1994, 343: 777--801

[28] Zhang G. Intersection bodies and the four-dimensional Busemann-Petty problem. Int Math Res Notices, 1993, 7: 233--240

[29] Haberl C, Ludwig M. A characterization of L_p-intersection bodies. Int Math Res Notices, Art ID, 2006, 10548: 29

[30] Schneider R. Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Cambridge Univ Press, 1993

[31] Gardner R J. Geometric Tomography. Cambridge: Cambridge Univ Press, 1995

[32] Lutwak E. Intersection bodies and dual mixed volumes. Adv Math, 1988, 71: 232--261

[33] Paouris G. ψ₂-estimates for linear functionals on zonoids, Geometric Aspects of Functional Analysis. (Milman-Schechtman eds), Lecture Notes in Math, 2003, 1807: 211--222

[34] Gardner R J, Giannopoulos A A. p-Cross-section bodies. Indiana Univ Math J, 1999, 48: 593--613

[35] Busemann H, Petty C M. Problem on convex bodies. Math Scand, 1956, 4: 88--94

[36] Lutwak E, Yang D, Zhang G Y. Sharp affine isoperimetric inequalities. J Differential Geom, 2000, 56: 111--132

[37] Yaskin V, Yaskina M. Centroid bodies and comparison of volumes. Indiana University Math J, 2006, 55(3): 1175--1194

[38] Lutwak E, Zhang G Y. Blaschke-Santal\'{o} inequalities. J Differential Geom, 1997, 47: 1--16

[39] Lutwak E. Dual mixed volumes. Pacific J Math, 1975, 58: 531--538

[40] 马统一. 混合投影体的极与混合相交体的Aleksandrov-Fenchel不等式的稳定性. 数学物理学报, 2009, 29A(6): 1750--1764

[41] Zhu Xianyang, Shen Yajun. The dual Brunn-Minkowski inequalities for star dual of mixed intersection bodies. Acta Mathematica Scientia, 2010, 30B(3): 739--746

[42] Lutwak E. Inequalities for mixed projection bodies. Trans Amer Math Soc, 1993, 339: 901--916

[43] Lutwak E. The Brunm-Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J Differential Geom, 1993, 38: 131--150