A new affine invariant for polytopes and schneider's projection problem

Transactions of the American Mathematical Society

Volumn 353, Issue 5, 2001, Pages 1767-1779

  Lutwak, Erwin ^a   Yang, Deane ^a   Zhang, Gaoyong ^a  

^a POLYTECHNIC UNIVERSITY   (United States)

Author keywords

Affine isoperimetric inequalities; Asymptotic inequalities; Projection bodies; Reverse isoperimetric inequalities

Indexed keywords

EID: 23044525631     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-01-02726-x     Document Type: Article

References (29)

1. [B] K. Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. 44 (1991), 351-359.MR 92j:52013
2. [BoLi] J. Bourgain and J. Lindcnstrauss, Projection bodies, Geometric Aspects of Functiona Analysis (J. Lindenstrauss and V.D. Milman, Eds.) Springer Lecture Notes in Math 1317 (1988), 250-270.MR 89g:46024
3. [BoM] J. Bourgain and V. Milman, New volume ratio properties for convex symmetric bodies i Rn, Invent. Math. 88 (1987), 319-340.MR 88f:52013
4. [Br] N.S. Brannen, Volumes of projection bodies, Mathematika 43 (1996), 255-264. M 98e:52005
5. [G] R. J. Gardner, Geometric Tomography, Cambridge Univ. Press, Cambridge, 1995. M 96j:52006
6. [GiPa] A. Giannopoulos and M. Papadimitrakis, Isotropie surface area measures, Mathematik 46 (1999), 1-14. CMP 2000:10
7. [GoW] P. R. Goodcy and W. Weil, Zonoids and generalizations, Handbook of Convex Geometr (P.M. Gruber and J.M. Wills, Eds.), North-Holland, Amsterdam, 1993, pp. 1297-1326 MR 95g:52015
8. [GMR] Y. Gordon, M. Meyer, and S. Reisner, Zonoids with minimal volume-product. A ne proof, Proc. Amor. Math. Soc. 104 (1988), 273-276.MR 89i:52015
9. [Le] K. Leichtwei, Affine Geometry of Convex Bodies, J.A. Barth, Heidelberg, 1998. M 2000j:52005
10. [LiM] J. Lindenstrauss and V. D. Milman, Local theory of normed spaces and convexity, Handbook of Convex Geometry (P.M. Gruber and J.M. Wills, Eds.), North-Holland, Amsterdam, 1993, pp. 1149-1220.MR 95b:46012
11. [L] E. Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. 60 (1990) 365-391.MR 90k:52024
12. [LZ] E. Lutwak and G. Zhang, Blaschke-Santalö inequalities, J. Differential Geom. 47 (1997) 1-16.MR 2000c:52011
13. [LYZ] E. Lutwak, D. Yang and G. Zhang, A new ellipsoid associated with convex bodies, Duk Math. J. 104 (2000), 375-390. CMP 2001:01
14. [Ma] G. Matheron, Random sets and integral geometry, Wiley, New York, 1975.MR 52:6828
15. [Me] P. McMullen, Non-linear angle-sum relations for polyhedral cones and polytopes, Math Proc. Camb. Phil. Soc. 78 (1975), 247-261.MR 52:15238
16. [MPal] V. D. Milman and A. Pajor, Cas limites des inégalités du type Khinchine et application géométriques, C.R. Acad. Sei. Paris 308 (1989), 91-96.MR 90d:52018
17. [MPa2] V. D. Milman and A. Pajor, Isotropie position and inertia ellipsoids and zonoids of th unit ball of a normed n-dimensional space, Geometric Aspects of Functional Analysis Springer Lecture Notes in Math. 1376 (1989), 64-104.MR 90g:52003
18. [PI] C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535-1547.MR 24:A3558
19. [P2] C. M. Petty, Projection bodies, Proc. Coll. Convexity, Copenhagen, 1965, Kpbenhavn Univ. Mat. Inst., 1967, pp. 234-241.MR 35:7203
20. [P3] C. M. Petty, Isoperimetric problems, Proc. Conf. Convexity and Combinatorial Geometr (Univ. Oklahoma, 1971), University of Oklahoma, 1972, pp. 26-41.MR 50:14499
21. [Rl] S. Reisner, Random polytopes and the volume-product of symmetric convex bodies, Math Scand. 57 (1985), 386-392.MR 87g:52011
22. [R2] S. Reisner, Zonoids with minimal volume-product, Math. Z. 192 (1986), 339-346. M 89i:52015
23. [SI] R. Schneider, Random hyperplanes meeting a convex body, Z. Wahrscheinlichkeitsth. vcrw Gcb. 61 (1982), 379-387.MR 84c:60023
24. [S2] R. Schneider, Convex Bodies: the Brunn-Alinkowski Theory, Cambridge Univ. Press Cambridge, 1993.MR 94d:52007
25. [SW] R. Schneider and W. Weil, Zonoids and related topics, Convexity and its Application (P.M. Gruber and J.M. Wills, Eds.), Birkhäuser, Basel, 1983, pp. 296-317.MR 85e:52010
26. [Sh] G. C. Shephard, Combinatorial properties of associated zonotopes, Canad. J. Math. 24 (1974), 302-321.MR 50:14496
27. [T] A.C. Thompson, Alinkowski Geometry, Cambridge Univ. Press, Cambridge, 1996. M 97f:52007
28. [W] W. Weil, Kontinuierliche Linearkombination von Strecken, Math. Z. 148 (1976), 71-84 MR 53:3887
29. [Z] G. Zhang, Restricted chord projection and affine inequalities, Geom. Dedicata 39 (1991) 213-222.MR 92f:52017