GL(n) equivariant minkowski valuations

Indiana University Mathematics Journal

Volumn 60, Issue 5, 2011, Pages 1655-1672

  Wannerer, Thomas ^a  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

Author keywords

Centroid body; Difference body; Valuation

Indexed keywords

EID: 84871498870     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2011.60.4425     Document Type: Article

References (42)

1. S. ALESKER, Continuous rotation invariant valuations on convex sets, Ann. of Math. (2) 149 (1999), no. 3, 977-1005. http://dx.doi.org/10.2307/121078. MR1709308 (2000i:52019)
2. Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture, Geom. Funct. Anal. 11 (2001), no. 2, 244-272. http://dx.doi.org/10.1007/PL00001675. MR1837364 (2002e:52015)
3. A. BERNIG, Valuations with Crofton formula and Finsler geometry, Adv. Math. 210 (2007), no. 2, 733-753. http://dx.doi.org/10.1016/j.aim.2006.07.009. MR2303237 (2008i:53108)
4. A. BERNIG and L. BR̈O CKER, Valuations on manifolds and Rumin cohomology, J. Differential Geom. 75 (2007), no. 3, 433-457. MR2301452 (2008e:53146)
5. A. BERNIG and J.H.G. FU, Hermitian integral geometry, Ann. of Math. (2) 173 (2011), no. 2, 907-945. http://dx.doi.org/10.4007/annals.2011.173.2.7. MR2301452
6. G. BERCK, Convexity of Lp-intersection bodies, Adv. Math. 222 (2009), no. 3, 920-936. http://dx.doi.org/10.1016/j.aim.2009.05.009. MR2553373 (2010m:52008)
7. S. CAMPI and P. GRONCHI, The Lp-Busemann-Petty centroid inequality, Adv.Math. 167 (2002), no. 1, 128-141. http://dx.doi.org/10.1006/aima.2001.2036. MR1901248 (2003e:52011)
8. B. FLEURY, O. GÚEDON, and G. PAOURIS, A stability result for mean width of Lp-centroid bodies, Adv. Math. 214 (2007), no. 2, 865-877. http://dx.doi.org/10.1016/j.aim.2007.03.008. MR2349721 (2008g:52012)
9. J.H.G. FU, Structure of the unitary valuation algebra, J. Differential Geom. 72 (2006), no. 3, 509-533. MR2219942 (2007b:52008) (Pubitemid 43781204)
10. R.J. GARDNER, Geometric Tomography, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 2006. MR2251886 (2007i:52010)
11. R.J. GARDNER and A.A. GIANNOPOULOS, p-cross-section bodies, Indiana Univ. Math. J. 48 (1999), no. 2, 593-613. http://dx.doi.org/10.1512/iumj.1999.48. 1689. MR1722809 (2000i:52002)
12. E. GRINBERG and G. ZHANG, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. (3) 78 (1999), no. 1, 77-115. http://dx.doi.org/10.1112/ S0024611599001653. MR1658156 (99m:52009)
13. C. HABERL, Lp intersection bodies, Adv. Math. 217 (2008), no. 6, 2599-2624. http://dx.doi.org/10.1016/j.aim.2007.11.013. MR2397461 (2009a:52001)
14. Star body valued valuations, Indiana Univ. Math. J. 58 (2009), no. 5, 2253-2276. http://dx.doi.org/10.1512/iumj.2009.58.3685. MR2583498 (2011b:52018)
15. Blaschke valuations, Amer. J. Math., in press.
16. C. HABERL and M. LUDWIG, A characterization of Lp intersection bodies, Int. Math. Res. Not. (2006), Art. ID 10548, 29. http://dx.doi.org/10.1155/IMRN/ 2006/10548. MR2250020 (2007k:52007)
17. C. HABERL and F.E. SCHUSTER, General Lp affine isoperimetric inequalities, J. Differential Geom. 83 (2009), no. 1, 1-26. MR2545028 (2010j:52026)
18. Asymmetric affine Lp Sobolev inequalities, J. Funct. Anal. 257 (2009), no. 3, 641-658. http://dx.doi.org/10.1016/j.jfa.2009.04.009. MR2530600 (2010j:46068)
19. M. KIDERLEN, Blaschke- and Minkowski-endomorphisms of convex bodies, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5539-5564 (electronic). http://dx.doi.org/10.1090/S0002-9947-06-03914-6. MR2238926 (2007e:52006) (Pubitemid 44972099)
20. D.A. KLAIN and G.-C. ROTA, Introduction to Geometric Probability, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1997. MR1608265 (2001f:52009)
21. M. LUDWIG, Projection bodies and valuations, Adv. Math. 172 (2002), no. 2, 158-168. http://dx.doi.org/10.1016/S0001-8708(02)00021-X. MR1942402 (2003j:52012) (Pubitemid 36100501)
22. Ellipsoids and matrix-valued valuations, Duke Math. J. 119 (2003), no. 1, 159-188. http://dx.doi.org/10.1215/S0012-7094-03-11915-8. MR1991649 (2004e:52015) (Pubitemid 36989960)
23. Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4191-4213 (electronic). http://dx.doi.org/10.1090/S0002-9947-04-03666-9. MR2159706 (2006f:52005) (Pubitemid 41421401)
24. Intersection bodies and valuations, Amer. J. Math. 128 (2006), no. 6, 1409-1428. MR2275906 (2008a:52012)
25. M. LUDWIG and M. REITZNER, A classification of SL(n) invariant valuations, Ann. of Math. (2) 172 (2010), no. 2, 1219-1267. http://dx.doi.org/ 10.4007/annals.2010.172.1223. MR2680490 (2011g:52025)
26. E. LUTWAK, On some affine isoperimetric inequalities, J. Differential Geom. 23 (1986), no. 1, 1-13. MR840399 (87k:52030)
27. Centroid bodies and dual mixed volumes, Proc. London Math. Soc. (3) 60 (1990), no. 2, 365-391. http://dx.doi.org/10.1112/plms/s3-60.2.365. MR1031458 (90k:52024)
28. E. LUTWAK and G. ZHANG, Blaschke-Santaĺo inequalities, J. Differential Geom. 47 (1997), no. 1, 1-16. MR1601426 (2000c:52011)
29. E. LUTWAK, D. YANG, and G. ZHANG, Lp affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111-132. MR1863023 (2002h:52011)
30. A new ellipsoid associated with convex bodies, DukeMath. J. 104 (2000), no. 3, 375-390. http://dx.doi.org/10.1215/S0012-7094-00-10432-2. MR1781476 (2001j:52011)
31. Orlicz centroid bodies, J. Differential Geom. 84 (2010), no. 2, 365-387. MR2652465
32. P. MCMULLEN, Valuations and dissections, Handbook of Convex Geometry, Vol. B (P.M. Gruber and J.M. Wills, eds.), North-Holland, Amsterdam, 1993, pp. 933-988. MR1243000 (95f:52018)
33. P. MCMULLEN and R. SCHNEIDER, Valuations on convex bodies, Convexity and Its Applications (P.M. Gruber and J.M. Wills, eds.), Birkḧauser, Basel, 1983, pp. 170-247. MR731112 (85e:52001)
34. G. PAOURIS, Concentration of mass on convex bodies, Geom. Funct. Anal. 16 (2006), no. 5, 1021-1049. http://dx.doi.org/10.1007/s00039-006-0584-5. MR2276533 (2007k:52009)
35. C.M. PETTY, Centroid surfaces, Pacific J. Math. 11 (1961), 1535-1547. MR0133733 (24 #A3558)
36. R. SCHNEIDER, Equivariant endomorphisms of the space of convex bodies, Trans. Amer.Math. Soc. 194 (1974), 53-78. MR0353147 (50 #5633)
37. Convex Bodies: the Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. http://dx.doi.org/10.1017/CBO9780511526282. MR1216521 (94d:52007)
38. R. SCHNEIDER and F.E. SCHUSTER, Rotation equivariant Minkowski valuations, Int. Math. Res. Not. (2006), Art. ID 72894, 20. http://dx.doi.org/ 10.1155/IMRN/2006/72894. MR2272092 (2008b:52009)
39. F.E. SCHUSTER, Convolutions and multiplier transformations of convex bodies, Trans. Amer.Math. Soc. 359 (2007), no. 11, 5567-5591. http://dx.doi.org/10.1090/S0002-9947-07-04270-5. MR2327043 (2009e:52008)
40. Crofton measures and Minkowski valuations, Duke Math. J. 154 (2010), no. 1, 1-30. http://dx.doi.org/10.1215/00127094-2010-033. MR2668553
41. F.E. SCHUSTER and T. WANNERER, GL(n) contravariant Minkowski valuations, Trans. Amer. Math. Soc., in press.
42. V. YASKIN and M. YASKINA, Centroid bodies and comparison of volumes, Indiana Univ. Math. J. 55 (2006), no. 3, 1175-1194. http://dx.doi.org/10.1512/ iumj.2006.55.2761. MR2244603 (2007f:52008) (Pubitemid 44214082)