The Lp-Minkowski problem and the Minkowski problem in centroaffine geometry

Advances in Mathematics

Volumn 205, Issue 1, 2006, Pages 33-83

  Chou, Kai Seng ^a   Wang, Xu Jia ^b  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

Blaschke Santalo inequality; Centroaffine Minkowski problem; Lp Minkowski problem; Monge Amp re equation

Indexed keywords

EID: 33746888293     PISSN: 00018708     EISSN: 10902082     Source Type: Journal    
DOI: 10.1016/j.aim.2005.07.004     Document Type: Article

References (24)

1. Ai J., Chou K.S., and Wei J. Self-similar solutions for the anisotropic affine curve shortening problem. Calc. Var. Partial Differential Equations 13 (2001) 311-337
2. Andrews B. Evolving convex curves. Cal. Var. Partial Differential Equations 7 (1998) 315-371
3. Andrews B. Gauss curvature flows: the fate of the rolling stones. Invent. Math. 138 (1999) 151-161
4. Andrews B. Motion of hypersurfaces by Gauss curvature. Pacific J. Math. 195 (2000) 1-34
5. Andrews B. Classification of limiting shapes for isotropic curve flows. J. Amer. Math. Soc. 16 (2003) 443-459
6. Caffarelli L.A. A localization property of viscosity solutions to the Monge-Ampire equation and their strict convexity. Ann. Math. 131 (1990) 129-134
7. 2, p-estimate for solutions of the Monge-Ampère equations. Ann. Math. 131 (1990) 135-150
8. K.S. Chou, X.J. Wang, Existence of solutions to a generalized Minkowski problem, Manuscript, 1995.
9. Chou K.S., and Wang X.J. The logarithmic Gauss curvature flow. Ann. Inst. H. Poincarè Anal. Non Linèaire 17 (2000) 733-751
10. Chou K.S., and Wang X.J. A variational theory of the Hessian equation. Comm. Pure Appl. Math. 54 (2001) 1029-1064
11. Cheng S.Y., and Yau S.T. On the regularity of the n-dimensional Minkowski problem. Comm. Pure Appl. Math. 20 (1977) 41-68
12. Firey W.J. p-means of convex bodies. Math. Scand. 10 (1968) 17-24
13. n, Manuscript, 1999.
14. C. Loewner, L. Nirenberg, Partial differential equations invariant under conformal or projective transformations, Contribution to Analysis, Academic Press, New York, 1974, pp. 245-272.
15. Lutwak E. The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J. Differential Geom. 38 (1993) 131-150 II: affine and geominimal surface areas, Adv. Math. 118 (1996) 244-294
16. Lutwak E., and Oliker V. On the regularity of solution to a generalization of the Minkowski problem. J. Differential Geom. 40 (1995) 227-246
17. p-Minkowski problem. Trans. Amer. Math. Soc. 356 (2004) 4359-4370
18. p Sobolev inequalities. J. Differential Geom. 62 (2002) 17-38
19. K. Nomizu, T. Sasaki, Affine Differential Geometry, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, New York, 1994.
20. Pogorelov A.V. The Multidimensional Minkowski Problem (1978), Wiley, New York
21. R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993.
22. Tzitzéica G. Sur une nouvelle classe de surfaces. Rend. Circ. Mat. Palermo 25 (1908) 180-187 28 (1909) 210-216
23. Urbas J. Complete noncompact self-similar solutions of Gauss curvature flows. I. Positive powers. Math. Ann. 311 (1998) 251-274
24. Urbas J. Complete noncompact self-similar solutions of Gauss curvature flows. II. Negative powers. Adv. Differential Equations 4 (1999) 323-346