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http://dx.doi.org/10.4134/BKMS.2010.47.5.1011

FOURIER TRANSFORM AND Lp-MIXED PROJECTION BODIES  

Liu, Lijuan (SCHOOL OF MATHEMATICS AND COMPUTATIONAL SCIENCE HUNAN UNIVERSITY OF SCIENCE AND TECHNOLOGY)
Wang, Wei (SCHOOL OF MATHEMATICS AND COMPUTATIONAL SCIENCE HUNAN UNIVERSITY OF SCIENCE AND TECHNOLOGY)
He, Binwu (DEPARTMENT OF MATHEMATICS SHANGHAI UNIVERSITY)
Publication Information
Bulletin of the Korean Mathematical Society / v.47, no.5, 2010 , pp. 1011-1023 More about this Journal
Abstract
In this paper we define the $L_p$-mixed curvature function of a convex body. We develop a formula connection the support function of $L_p$-mixed projection body with Fourier transform of the $L_p$-mixed curvature function. Using this formula we solve an analog of the Shephard projection problem for $L_p$-mixed projection bodies.
Keywords
Fourier transform; $L_p$-mixed curvature function; $L_p$-mixed projection body;
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1 E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244-294.   DOI   ScienceOn
2 E. Lutwak, D. Yang, and G. Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111-132.   DOI
3 S. J. Lv, On an analytic generalization of the Busemann-Petty problem, J. Math. Anal. Appl. 341 (2008), no. 2, 1438-1444.   DOI   ScienceOn
4 C. M. Petty, Projection bodies, Proc. Colloquium on Convexity (Copenhagen, 1965) pp. 234-241 Kobenhavns Univ. Mat. Inst., Copenhagen, 1967.
5 D. Ryabogin and A. Zvavitch, The Fourier transform and Firey projections of convex bodies, Indiana Univ. Math. J. 53 (2004), no. 3, 667-682.   DOI
6 R. Schneider, Zur einem Problem von Shephard uber die Projektionen konvexer Korper, Math. Z. 101 (1967), 71-82.   DOI
7 R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993.
8 A. Zvavitch, The Busemann-Petty problem for arbitrary measures, Math. Ann. 331 (2005), no. 4, 867-887.   DOI
9 R. J. Gardner, Geometric Tomography, Second edition. Encyclopedia of Mathematics and its Applications, 58. Cambridge University Press, Cambridge, 2006.
10 R. J. Gardner, A. Koldobsky, and Th. Schlumprecht, An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. of Math. (2) 149 (1999), no. 2, 691-703.   DOI
11 I. M. Gelfand and G. E. Shilov, Generalized Functions. Vol. 1, Properties and operations. Academic Press, New York-London, 1964.
12 I. M. Gelfand and N. Y. Vilenkin, Generalized Functions, Academic Press, New York-London, 1964.
13 A. Koldobsky, Intersection bodies, positive definite distributions, and the Busemann-Petty problem, Amer. J. Math. 120 (1998), no. 4, 827-840.   DOI
14 A. Koldobsky, A generalization of the Busemann-Petty problem on sections of convex bodies, Israel J. Math. 110 (1999), 75-91.   DOI
15 E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131-150.   DOI
16 A. Koldobsky, Fourier Analysis in Convex Geometry, Mathematical Surveys and Monographs, 116. American Mathematical Society, Providence, RI, 2005.
17 A. Koldobsky, D. Ryabogin, and A. Zvavitch, Projections of convex bodies and the Fourier transform, Israel J. Math. 139 (2004), 361-380.   DOI