FOURIER TRANSFORM AND Lp-MIXED PROJECTION BODIES

Title & Authors
FOURIER TRANSFORM AND Lp-MIXED PROJECTION BODIES
Liu, Lijuan; Wang, Wei; He, Binwu;

Abstract
In this paper we define the $\small{L_p}$-mixed curvature function of a convex body. We develop a formula connection the support function of $\small{L_p}$-mixed projection body with Fourier transform of the $\small{L_p}$-mixed curvature function. Using this formula we solve an analog of the Shephard projection problem for $\small{L_p}$-mixed projection bodies.
Keywords
Fourier transform;$\small{L_p}$-mixed curvature function;$\small{L_p}$-mixed projection body;
Language
English
Cited by
1.
The ith p-affine surface area, Journal of Inequalities and Applications, 2015, 2015, 1
2.
Some inequalities related to (i,j)-type Lp-mixed affine surface area and Lp-mixed curvature image, Journal of Inequalities and Applications, 2013, 2013, 1, 470
3.
The generalized L p -mixed affine surface area, Acta Mathematica Sinica, English Series, 2015, 31, 11, 1775
4.
The ith p-geominimal surface area, Journal of Inequalities and Applications, 2014, 2014, 1, 356
5.
Stability in the shephard problem for L p -projection of convex bodies, Wuhan University Journal of Natural Sciences, 2014, 19, 4, 283
References
1.
R. J. Gardner, Geometric Tomography, Second edition. Encyclopedia of Mathematics and its Applications, 58. Cambridge University Press, Cambridge, 2006.

2.
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.

3.
I. M. Gelfand and G. E. Shilov, Generalized Functions. Vol. 1, Properties and operations. Academic Press, New York-London, 1964.

4.
I. M. Gelfand and N. Y. Vilenkin, Generalized Functions, Academic Press, New York-London, 1964.

5.
A. Koldobsky, Intersection bodies, positive definite distributions, and the Busemann-Petty problem, Amer. J. Math. 120 (1998), no. 4, 827-840.

6.
A. Koldobsky, A generalization of the Busemann-Petty problem on sections of convex bodies, Israel J. Math. 110 (1999), 75-91.

7.
A. Koldobsky, Fourier Analysis in Convex Geometry, Mathematical Surveys and Monographs, 116. American Mathematical Society, Providence, RI, 2005.

8.
A. Koldobsky, D. Ryabogin, and A. Zvavitch, Projections of convex bodies and the Fourier transform, Israel J. Math. 139 (2004), 361-380.

9.
E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131-150.

10.
E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244-294.

11.
E. Lutwak, D. Yang, and G. Zhang, \$L_p\$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111-132.

12.
S. J. Lv, On an analytic generalization of the Busemann-Petty problem, J. Math. Anal. Appl. 341 (2008), no. 2, 1438-1444.

13.
C. M. Petty, Projection bodies, Proc. Colloquium on Convexity (Copenhagen, 1965) pp. 234-241 Kobenhavns Univ. Mat. Inst., Copenhagen, 1967.

14.
D. Ryabogin and A. Zvavitch, The Fourier transform and Firey projections of convex bodies, Indiana Univ. Math. J. 53 (2004), no. 3, 667-682.

15.
R. Schneider, Zur einem Problem von Shephard uber die Projektionen konvexer Korper, Math. Z. 101 (1967), 71-82.

16.
R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993.

17.
A. Zvavitch, The Busemann-Petty problem for arbitrary measures, Math. Ann. 331 (2005), no. 4, 867-887.