Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Theory of valuations on manifolds, III. Multiplicative structure in the general case

Authors: Semyon Alesker and Joseph H. G. Fu
Journal: Trans. Amer. Math. Soc. 360 (2008), 1951-1981
MSC (2000): Primary 52B45, 52A39, 53C65
Published electronically: November 27, 2007
MathSciNet review: 2366970
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This is the third part of a series of articles where the theory of valuations on manifolds is constructed. In the second part of this series the notion of a smooth valuation on a manifold was introduced. The goal of this article is to put a canonical multiplicative structure on the space of smooth valuations on general manifolds, thus extending some of the affine constructions from the first author's 2004 paper and, from the first part of this series.

References [Enhancements On Off] (What's this?)

  • 1. Alesker, Semyon; Integrals of smooth and analytic functions over Minkowski's sums of convex sets. MSRI ``Convex Geometric Analysis'' 34 (1998), 1-15. MR 1665573 (99m:52006)
  • 2. Alesker, Semyon; Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture. Geom. Funct. Anal. 11 (2001), no. 2, 244-272. MR 1837364 (2002e:52015)
  • 3. Alesker, Semyon; The multiplicative structure on polynomial continuous valuations. Geom. Funct. Anal. 14 (2004), no. 1, 1-26, also: math.MG/0301148. MR 2053598 (2005d:52022)
  • 4. Alesker, Semyon; Theory of valuations on manifolds, I. Linear spaces. Israel J. Math., 156 (2006), 311-339. MR 2282381
  • 5. Alesker, Semyon; Theory of valuations on manifolds, II. Adv. Math., 2007 (2006), 420-454. MR 2264077
  • 6. Alesker, Semyon; Theory of valuations on manifolds, IV. New properties of the multiplicative structure. math.MG/0511171.
  • 7. Chern, S. S.; Curves and surfaces in Euclidean space. 1967 Studies in Global Geometry and Analysis, pp. 16-56. Math. Assoc. Amer. MR 0212744 (35:3610)
  • 8. Federer, Herbert; Curvature measures. Trans. Amer. Math. Soc. 93 1959 418-491. MR 0110078 (22:961)
  • 9. Federer, Herbert; Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band 153 Springer-Verlag New York Inc., New York 1969. MR 0257325 (41:1976)
  • 10. Fu, Joseph H. G.; Monge-Ampère functions. I. Indiana Univ. Math. J. 38 (1989), no. 3, 745-771. MR 1017333 (91d:49048)
  • 11. Fu, Joseph H. G.; Kinematic formulas in integral geometry. Indiana Univ. Math. J. 39 (1990), no. 4, 1115-1154. MR 1087187 (92c:53043)
  • 12. Fu, Joseph H. G.; Convergence of curvatures in secant approximations. J. Differential Geom. 37 (1993), no. 1, 177-190. MR 1198604 (94a:53103)
  • 13. Fu, Joseph H. G.; Curvature measures of subanalytic sets. Amer. J. Math. 116 (1994), no. 4, 819-880. MR 1287941 (95g:32016)
  • 14. Harvey, Reese; Lawson, H. Blaine, Jr.; Calibrated geometries. Acta Math. 148 (1982), 47-157. MR 666108 (85i:53058)
  • 15. Kashiwara, Masaki; Schapira, Pierre; Sheaves on manifolds. With a chapter in French by Christian Houzel. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 292. Springer-Verlag, Berlin, 1990. MR 1074006 (92a:58132)
  • 16. McMullen, Peter; Valuations and dissections. Handbook of convex geometry, Vol. A, B, 933-988, North-Holland, Amsterdam, 1993. MR 1243000 (95f:52018)
  • 17. McMullen, Peter; Schneider, Rolf; Valuations on convex bodies. Convexity and its applications, 170-247, Birkhäuser, Basel, 1983. MR 731112 (85e:52001)
  • 18. Schneider, Rolf; Convex bodies: the Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1993. MR 1216521 (94d:52007)
  • 19. White, Brian; A new proof of the compactness theorem for integral currents. Comment. Math. Helv. 64 (1989), no. 2, 207-220. MR 997362 (90e:49052)
  • 20. Whitney, Hassler; Geometric integration theory. Princeton University Press, Princeton, N. J., 1957. MR 0087148 (19:309c)
  • 21. Zähle, Martina; Approximation and characterization of generalised Lipschitz-Killing curvatures. Ann. Global Anal. Geom. 8 (1990), no. 3, 249-260. MR 1089237 (91m:53055)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 52B45, 52A39, 53C65

Retrieve articles in all journals with MSC (2000): 52B45, 52A39, 53C65

Additional Information

Semyon Alesker
Affiliation: Department of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel

Joseph H. G. Fu
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Received by editor(s): October 21, 2005
Received by editor(s) in revised form: December 2, 2005
Published electronically: November 27, 2007
Additional Notes: The first author was partially supported by ISF grant 1369/04.
The second author was partially supported by NSF grant DMS-0204826.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society