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Theory of valuations on manifolds, I. Linear spaces

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Abstract

This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. We introduce a space of smooth (non-translation invariant) valuations on a linear spaceV. We present three descriptions of this space. We describe the canonical multiplicative structure on this space generalizing the results from [4] obtained for polynomial valuations.

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References

  1. S. Alesker,Integrals of smooth and analytic functions over Minkowski's sums of convex sets, MSRI “Convex Geometric Analysis”34 (1998), 1–15.

    MathSciNet  Google Scholar 

  2. S. Alesker,On P. McMullen's conjecture on translation invariant valuations, Advances in Mathematics155 (2000), 239–263.

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Alesker,Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture, Geometric and Functional Analysis11 (2001), 244–272.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Alesker,The multiplicative structure on polynomial continuous valuations, Geometric and Functional Analysis14 (2004), 1–26; also: math.MG/0301148.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Alesker,Theory of valuations on manifolds, II, Advances in Mathematics, to appear; math.MG/0503399.

  6. S. Alesker and J. H. G. Fu,Theory of valuations on manifolds, III. Multiplicative structure in the general case, Transactions of the Mathematical American Society, to appear; also: math.MG/0509512.

  7. W. Casselman,Canonical extensions of Harish-Chandra modules to representations of G, Canadian Journal of Mathematics41 (1989), 385–438.

    MATH  MathSciNet  Google Scholar 

  8. J. H. G. Fu,Curvature measures of subanalytic sets, American Journal of Mathematics116 (1994), 819–880.

    Article  MATH  MathSciNet  Google Scholar 

  9. I. M. Gelfand and N. Ya. Vilenkin,Generalized Functions, Vol. 4.Applications of Harmonic Analysis, Translated from the Russian by Amiel Feinstein, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977].

    Google Scholar 

  10. R. Godement,Topologie algébrique et théorie des faisceaux, (French) Troisième édition revue et corrigée, Publications de l'Institut de Mathématique de l'Université de Strasbourg, XIII. Actualités Scientifiques et Industrielles, No. 1252, Hermann, Paris, 1973.

    Google Scholar 

  11. H. Hadwiger,Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957.

    MATH  Google Scholar 

  12. M. Kashiwara and P. Schapira,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.

    MATH  Google Scholar 

  13. A. G. Khovanskii and A. V. Pukhlikov,Finitely additive measures of virtual polyhedra, (Russian) Algebra i Analiz4 (1992), no. 2, 161–185; translation in St. Petersburg Mathematical Journal4 (1993), no. 2, 337–356.

    MathSciNet  Google Scholar 

  14. P. McMullen,Valuations and Euler-type relations on certain classes of convex polytopes, Proceedings of the London Mathematical Society (3)35 (1977), 113–135.

    Article  MATH  MathSciNet  Google Scholar 

  15. P. McMullen,Valuations and dissections, inHandbook of Convex Geometry (P. M. Gruber and J. M. Willis, eds.), Vol. A, B, North-Holland, Amsterdam, 1993, pp. 933–988.

    Google Scholar 

  16. P. McMullen and R. Schneider,Valuations on convex bodies, inConvexity and its Applications (P. M. Gruber and J. M. Willis, eds.), Birkhäuser, Basel, 1983, pp. 170–247.

    Google Scholar 

  17. R. Schneider,Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993.

    MATH  Google Scholar 

  18. N. R. Wallach,Real reductive groups. I, II, Pure and Applied Mathematics, 132, Academic Press, Boston, 1988, 1992.

    Google Scholar 

  19. P. Wintgen,Normal cycle and integral curvature for polyhedra in riemannian manifolds, inDifferential Geometry (G. Soos and J. Szenthe, eds.), North-Holland, Amsterdam, 1982.

    Google Scholar 

  20. M. Zähle,Curvatures and currents for unions of sets with positive reach, Geometriae Dedicata23 (1987), 155–171.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Tel Aviv University Ramat Aviv, 69978, Tel Aviv, Israel

    Semyon Alesker

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  1. Semyon Alesker
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Partially supported by ISF grant 1369/04.

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Alesker, S. Theory of valuations on manifolds, I. Linear spaces. Isr. J. Math. 156, 311–339 (2006). https://doi.org/10.1007/BF02773837

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  • DOI: https://doi.org/10.1007/BF02773837

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