New Lp affine isoperimetric inequalities

Advances in Mathematics

Volumn 218, Issue 3, 2008, Pages 762-780

  Werner, Elisabeth ^a,b   Ye, Deping ^a  

^a CASE WESTERN RESERVE UNIVERSITY   (United States)

^b UNIV LILLE   (France)

Author keywords

Affine surface area; Lp Brunn Minkowski theory

Indexed keywords

EID: 41849095075     PISSN: 00018708     EISSN: 10902082     Source Type: Journal    
DOI: 10.1016/j.aim.2008.02.002     Document Type: Article

References (43)

1. Alesker S. Continuous rotation invariant valuations on convex sets. Ann. of Math. 149 (1999) 977-1005
2. Alesker S. Description of translation invariant valuations on convex sets with a solution of P. McMullen's conjecture. Geom. Funct. Anal. 11 (2001) 244-272
3. Andrews B. Contraction of convex hypersurfaces by their affine normal. J. Differential Geom. 43 2 (1996) 207-230
4. Andrews B. Gauss curvature flow: The fate of the rolling stones. Invent. Math. 138 (1999) 151-161
5. I. Bárány, Random points, convex bodies, lattices, in: Proceedings International Congress of Mathematicians, vol. III, Beijing, 2002, pp. 527-536
6. Blaschke W. Vorlesungen über Differentialgeometrie II, Affine Differentialgeometrie (1923), Springer-Verlag, Berlin
7. b. Invent. Math. 88 (1987) 319-340
8. p Busemann-Petty centroid inequality. Adv. Math. 167 (2002) 128-141
9. p Minkowski problem with not necessarily positive data. Adv. Math. 201 (2006) 77-89
10. p-Minkowski problem and the Minkowski problem in centroaffine geometry. Adv. Math. 205 (2006) 33-83
11. p-centroid bodies. Adv. Math. 214 (2007) 865-877
12. Gardner R.J. Geometric Tomography (1995), Cambridge Univ. Press
13. Gardner R.J., and Zhang G. Affine inequalities and radial mean bodies. Amer. J. Math. 120 3 (1998) 505-528
14. Gruber P.M. Aspects of approximation of convex bodies. Handbook of Convex Geometry, vol. A (1993), North-Holland 321-345
15. Hug D. Curvature relations and affine surface area for a general convex body and its polar. Results Math. 29 (1996) 233-248
16. Hu C., Ma X., and Shen C. On the Christoffel-Minkowski problem of Firey's p-sum. Calc. Var. Partial Differential Equations 21 2 (2004) 137-155
17. Klain D. Invariant valuations on star shaped sets. Adv. Math. 125 (1997) 95-113
18. Klartag B. A central limit theorem for convex sets. arXiv: math.MG/0605014 V. 2, 23 Oct. 2006
19. G. Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal., in press
20. Leichtweiss K. Affine Geometry of Convex Bodies (1998), Johann Ambrosius Barth Verlag, Heidelberg
21. Ludwig M. Ellipsoids and matrix-valued valuations. Duke Math. J. 119 (2003) 159-188
22. Ludwig M., and Reitzner M. A characterization of affine surface area. Adv. Math. 147 (1999) 138-172
23. M. Ludwig, M. Reitzner, A classification of SL (n) invariant valuations, Ann. of Math., in press
24. Ludwig M., Schütt C., and Werner E. Approximation of the Euclidean ball by polytopes. Studia Math. 173 (2006) 1-18
25. Lutwak E. Extended affine surface area. Adv. Math. 85 (1991) 39-68
26. Lutwak E. The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas. Adv. Math. 118 2 (1996) 244-294
27. Lutwak E., and Oliker V. On the regularity of solutions to a generalization of the Minkowski problem. J. Differential Geom. 41 (1995) 227-246
28. Lutwak E., Yang D., and Zhang G. A new ellipsoid associated with convex bodies. Duke Math. J. 104 (2000) 375-390
29. p affine isoperimetric inequalities. J. Differential Geom. 56 (2000) 111-132
30. p Sobolev inequalities. J. Differential Geom. 62 (2002) 17-38
31. Meyer M., and Werner E. The Santaló-regions of a convex body. Trans. Amer. Math. Soc. 350 11 (1998) 4569-4591
32. Meyer M., and Werner E. On the p-affine surface area. Adv. Math. 152 (2000) 288-313
33. Sapiro G., and Tannenbaum A. On affine plane curve evolution. J. Funct. Anal. 119 (1994) 79-120
34. n. Israel J. Math. 78 (1992) 309-334
35. Schneider R. Convex Bodies: The Brunn-Minkowski Theory (1993), Cambridge Univ. Press
36. Schütt C., and Werner E. The convex floating body. Math. Scand. 66 (1990) 275-290
37. Schütt C., and Werner E. Random polytopes of points chosen from the boundary of a convex body. GAFA Seminar Notes. Lecture Notes in Math. vol. 1807 (2002), Springer-Verlag 241-422
38. Schütt C., and Werner E. Surface bodies and p-affine surface area. Adv. Math. 187 (2004) 98-145
39. 0-Minkowski problem. Adv. Math. 167 (2002) 160-174
40. 0-Minkowski problem. Adv. Math. 180 (2003) 290-323
41. Trudinger N.S., and Wang X. The affine Plateau problem. J. Amer. Math. Soc. 18 (2005) 253-289
42. X. Wang, Affine maximal hypersurfaces, in: Proceedings of the International Congress of Mathematicians, vol. III, Beijing, 2002, pp. 221-231
43. Werner E. Illumination bodies and affine surface area. Studia Math. 110 (1994) 257-269