一般测度Busemann-Petty问题的稳定性
Stability in the Busemann-Petty Problem for Arbitrary Measures
DOI: 10.12677/PM.2012.24034, PDF, HTML,    国家自然科学基金支持
作者: 汪 卫*：湖南科技大学数学与计算科学学院
关键词: Busemann-Petty问题；星体；凸体；Radon变换；The Busemann-Petty Problem; Star Bodies; Convex Bodies; Radon Transform
摘要: 基于Zvavitch将Busemann-Petty问题推广到了一般测度，本文利用Radon变换研究了一般测度Busemann-Petty问题的稳定性。作为应用，我们建立了n(n≤4)维空间中的一个关于一般测度的超截面不等式。这些结果与Koldobsky利用Fourier变换证明的结论是一致的。
Abstract: Zvavitch found a generalization of the Busemann-Petty problem to arbitrary measures. In this paper, we study the stability in the Busemann-Petty problem for arbitrary measures by using Radon transform. As application, we obtain a hyperplane inequality for arbitrary measures in dimensions up to four. These results are consistent with Koldobsky’s results which are obtained by using Fourier transform.
文章引用：汪卫. 一般测度Busemann-Petty问题的稳定性[J]. 理论数学, 2012, 2(4): 221-225. http://dx.doi.org/10.12677/PM.2012.24034

参考文献

[1] H. Busemann, C. M. Petty. Problem on convex bodies. Mathematica Scandinavica, 1956, 4: 88-94.
[2] D. G. Larman, C. A. Rogers. The existence of a centrally symmetric convex body with central sections that are unexpectedly small. Mathematika, 1975, 22(2): 164-175.
[3] K. Ball. Cube slicing in Rn. Proceedings of the American Mathematical Society, 1986, 97(3): 465-473.
[4] A. A. Giannopoulos. A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies. Mathematika, 1990, 37: 239-244.
[5] J. Bourgain. On the Busemann-Petty problem for perturbations of the ball. Geometric and Functional Analysis, 1991, 1(1): 1-13.
[6] M. Papadimitrakis. On the Busemann-Petty problem about convex, centrally symmetric bodies in Rn. Mathematika, 1992, 39: 258-266.
[7] R. J. Gardner. Intersection bodies and the Busemann-Petty problem. Transactions on American Mathematical Society, 1994, 342(1): 435-445.
[8] R. J. Gardner. A positive answer to the Busemann-Petty problem in three dimensions. Annals of Mathematics, 1994, 140(2): 435-447.
[9] G. Y. Zhang. A positive answer to the Busemann-Petty problem in four dimensions. Annals of Mathematics, 1999, 149: 535-543.
[10] R. J. Gardner, A. Koldobsky and Th. Schlumprecht. An analytic solution of the Busemann-Petty problem on sections of convex bodies. Annals of Mathematics, 1999, 149: 691-703.
[11] R. J. Gardner. Geometric tomography (2nd edition). New York: Cambridge University Press, 2006.
[12] A. Koldobsky. Fourier analysis in convex geometry. Mathematical Surveys and Monographs, Vol. 116, American Mathematical Society, 2005.
[13] A. Zvavitch. The Busemann-Petty problem for arbitrary measures. Mathematische Annalen, 2005, 331(4): 867-887.
[14] K. J. Böröczky. Stability of the Blaschke-Santaló and the affine isoperimetric inequality. Advances in Mathematics, 2010, 225(4): 1914-1928.
[15] K. J. Böröczky, D. Hug. Stability of the reverse Blaschke-Santaló inequality for zonoids and applications. Advances in Applied Mathematics, 2010, 44(4): 309-328.
[16] R. J. Gardner, S. Vassallo. Stability of inequalities in the dual Brunn-Minkowski theory. Journal of Mathematical Analysis and Application, 1999, 231(2): 568-587.
[17] A. Koldobsky. Stability in the Busemann-Petty and Shephard problems. Advances in Mathematics, 2011, 228(4): 2145-2161.
[18] A. Koldobsky. Stability of volume comparsion for complex convex bodies. Archiv der Mathematik, 2011, 97: 91-98.
[19] R. Schneider. Stability in the Aleksandrov-Fenchel-Jessen theorem. Mathematika, 1989, 36: 50-59.
[20] 何斌吾, 李小燕, 冷岗松. 对偶Aleksandrov-Fenchel不等式的稳定性[J]. 数学学报, 2005, 48: 1071-1078.
[21] P. Goodey, W. Weil. Centrally symmetric convex bodies and the spherical Radon transform. Journal of Differential Geometry, 1992, 35(3): 675-688.
[22] E. Lutwak. Intersection bodies and dual mixed volumes. Advances in Mathematics, 1988, 71(2): 232-261.
[23] P. Goodey, E. Lutwak and W. Weil. Functional analytic characterizations of classes of convex bodies. Mathematische Zeischrift, 1996, 222: 363-381.
[24] A. Koldobsky. A hyperplane inequality for measures of convex bodies in Rn, n 4. Discrete & Computational Geometry, 2012, 47(3): 538-547.
[25] B. Klartag. On convex perturbations with a bounded isotropic constant. Geometric and Functional Analysis, 2006, 16(6): 1274-1290.

为你推荐