The Fourier transform and Firey projections of convex bodies

Indiana University Mathematics Journal

Volumn 53, Issue 3, 2004, Pages 667-682

  Ryabogin, D ^a   Zvavitch, A ^b  

^a KANSAS STATE UNIVERSITY   (United States)

^b UNIVERSITY OF MISSOURI   (United States)

Author keywords

Brunn Minkowski Firey theory; Fourier transform; Parseval's identity; Projections; Shepard problem

Indexed keywords

EID: 4944246828     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2004.53.2399     Document Type: Article

References (16)

1. WM. J. FIREY, p-means of convex bodies, Math. Scand. 10 (1962), 17-24. MR 25 #4416
2. RICHARD J. GARDNER, Geometric Tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995, ISBN 0-521-45126-4. MR 96j:52006
3. R. J. GARDNER, A. KOLDOBSKY, and T. SCHLUMPRECHT, An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. of Math. (2) 149 (1999), 691-703. MR 2001b:52011
4. ERWIN LUTWAK, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), 232-261. MR 90a:52023
5. _, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131-150. MR 94g:52008
6. Erwin LUTWAK, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), 244-294. MR 97f:52014
7. p affine isoperimetric inequalities, J. Differential Geom. 56 (2000), 111-132. MR 2002h:52011
8. I. M. GEL'FAND and G. E. SHILOV, Generalized Functions. Vol. I: Properties and Operations, Translated by Eugene Saletan, Academic Press, New York, 1964. MR 29 #3869
9. ALEXANDER L. KOLDOBSKY, Generalized Lévy representation of norm and isometric embedding into Lp-spaces, Ann. Inst. H. Poincaré Probab. Statist. 28 (1992), 335-353. MR 93f:46040 (English, with English and French summaries)
10. ALEXANDER KOLDOBSKY, Inverse formula far the Blaschke-Levy representation, Houston J. Math. 23 (1997), 95-108. MR 2000b:42005
11. A. KOLDOBSKY, A generalization of the Busemann-Petty problem on sections of convex bodies, Israel J. Math. 110 (1999), 75-91. MR 2000m:52017
12. ALEXANDER KOLDOBSKY, DMITRY RYABOGIN, and ARTEM ZVAVITCH, Projections of convex bodies and the Fourier transform, Israel J. Math. 139 (2004), 361-380. 2 041 799
13. C. M. PETTY, Projection bodies, Proc. Colloquium on Convexity (Copenhagen, 1965), Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 234-241. MR 35 #7203
14. ROLF SCHNEIDER, Zur einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z. 101 (1967), 71-82. MR 36 #2059 (German)
15. _, Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993, ISBN 0-521-35220-7. MR 94d:52007
16. V.I. SEMYANISTYI, Some integral transformations and integral geometry in an elliptic space, Trudy Sem. Vector. Tenzor. Anal. 12 (1963), 297-411. (Russian)