Skip to main content

New Structures on Valuations and Applications

  • Chapter
  • First Online:
Integral Geometry and Valuations

Part of the book series: Advanced Courses in Mathematics - CRM Barcelona ((ACMBIRK))

Abstract

The theory of valuations on convex sets is a classical part of the topic of onvexity, with traditionally strong relations to integral geometry. During the roughly last 15 years a considerable progress was made in valuation theory and its applications to integral geometry.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

eBook
USD 24.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Alesker, S.: On P. McMullen’s conjecture on translation invariant valuations. Adv. Math. 155 (2000), no. 2, 239–263.

    MathSciNet  MATH  Google Scholar 

  2. Alesker, S.: Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture. Geom. Funct. Anal. 11 (2001), no. 2, 244–272.

    Article  MathSciNet  MATH  Google Scholar 

  3. Alesker, S.: Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations. J. Differential Geom. 63 (2003), no. 1, 63–95.

    MathSciNet  MATH  Google Scholar 

  4. Alesker, S.: The multiplicative structure on polynomial continuous valuations. Geom. Funct. Anal. 14 (2004), no. 1, 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  5. Alesker, S.: SU(2)-invariant valuations. In: Geometric Aspects of Functional Analysis, 21–29, Lecture Notes in Math. 1850, Springer, Berlin, 2004.

    Google Scholar 

  6. Alesker, S.: Hard Lefschetz theorem for valuations and related questions of integral geometry. In: Geometric Aspects of Functional Analysis, 9–20, Lecture Notes in Math. 1850, Springer, Berlin, 2004.

    Google Scholar 

  7. Alesker, S.: Valuations on convex sets, non-commutative determinants, and pluripotential theory. Adv. Math. 195 (2005), no. 2, 561–595.

    Article  MathSciNet  MATH  Google Scholar 

  8. Alesker, S.: Theory of valuations on manifolds, I. Linear spaces. Israel J. Math. 156 (2006), 311–339.

    Article  MathSciNet  MATH  Google Scholar 

  9. Alesker, S.: Theory of valuations on manifolds, II. Adv. Math. 207 (2006), no. 1, 420–454.

    Article  MathSciNet  MATH  Google Scholar 

  10. Alesker, S.: Theory of valuations on manifolds, IV. New properties of the multiplicative structure. In: Geometric Aspects of Functional Analysis, 1–44, Lecture Notes in Math. 1910, Springer, Berlin, 2007.

    Google Scholar 

  11. Alesker, S.: Theory of valuations on manifolds: a survey. Geom. Funct. Anal. 17 (2007), no. 4, 1321–1341.

    Article  MathSciNet  MATH  Google Scholar 

  12. Alesker, S.: Quaternionic plurisubharmonic functions and their applications to convexity. St. Petersburg Math. J. 19 (2008), no. 1, 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  13. Alesker, S.: Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets. J. Geom. Anal. 18 (2008), no. 3, 651–686.

    Article  MathSciNet  MATH  Google Scholar 

  14. Alesker, S.: Valuations on manifolds and integral geometry. Geom. Funct. Anal. 20 (2010), no. 5, 1073–1143.

    Article  MathSciNet  MATH  Google Scholar 

  15. Alesker, S.: A Fourier type transform on translation invariant valuations on convex sets. Israel J. Math. 181 (2011), 189–294.

    Article  MathSciNet  MATH  Google Scholar 

  16. Alesker, S. and Bernig, A.: The product on smooth and generalized valuations. Amer. J. Math. 134 (2012), no. 2, 507–560.

    Article  MathSciNet  MATH  Google Scholar 

  17. Alesker, S. and Bernstein, J.: Range characterization of the cosine transform on higher Grassmannians. Adv. Math. 184 (2004), no. 2, 367–379.

    Article  MathSciNet  MATH  Google Scholar 

  18. Alesker, S. and Fu, J.H.G.: Theory of valuations on manifolds, III. Multiplicative structure in the general case. Trans. Amer. Math. Soc. 360 (2008), no. 4, 1951–1981.

    MathSciNet  MATH  Google Scholar 

  19. Bernig, A.: A Hadwiger-type theorem for the special unitary group. Geom. Func. Anal. 19 (2009), 356–372.

    Article  MathSciNet  MATH  Google Scholar 

  20. Bernig, A.: Integral geometry under G2 and Spin(7). Israel J. Math. 184 (2011), 301–316.

    Article  MathSciNet  MATH  Google Scholar 

  21. Bernig, A.: Invariant valuations on quaternionic vector spaces. J. Inst. Math. Jussieu 11 (2012), 467–499.

    Article  MathSciNet  MATH  Google Scholar 

  22. Bernig, A.: Algebraic integral geometry. In: Global Differential Geometry (C. Bär, J. Lohkamp and M. Schwarz, eds.), 107–145, Springer Proceedings in Math. Vol. 17, Springer, Berlin, 2012.

    Google Scholar 

  23. Bernig, A. and Bröcker, L.: Valuations on manifolds and Rumin cohomology. J. Differential Geom. 75 (2007), 433–457.

    MathSciNet  MATH  Google Scholar 

  24. Bernig, A. and Fu, J.H.G.: Convolution of convex valuations. Geom. Dedicata 123 (2006), 153–169.

    Article  MathSciNet  MATH  Google Scholar 

  25. Bernig, A. and Fu, J.H.G.: Hermitian integral geometry. Ann. of Math. 173 (2011), 907–945.

    Article  MathSciNet  MATH  Google Scholar 

  26. Chern, S.-S.: On the curvatura integra in a Riemannian manifold. Ann. of Math. 46 (1945), 674–684.

    Article  MathSciNet  MATH  Google Scholar 

  27. Federer, H.: Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418–491.

    Article  MathSciNet  MATH  Google Scholar 

  28. Fu, J.H.G.: Curvature measures and generalized Morse theory, J. Differential Geom. 30 (1989), 619–642.

    MathSciNet  MATH  Google Scholar 

  29. Fu, J.H.G.: Monge–Ampère functions, I. Indiana Univ. Math. J. 38 (1989), no. 3, 745–771.

    Article  MATH  Google Scholar 

  30. Fu, J.H.G.: Monge–Ampère functions, II. Indiana Univ. Math. J. 38 (1989), no. 3, 773–789.

    Article  MATH  Google Scholar 

  31. Fu, J.H.G.: Kinematic formulas in integral geometry. Indiana Univ. Math. J. 39 (1990), no. 4, 1115–1154.

    Article  MathSciNet  MATH  Google Scholar 

  32. Fu, J.H.G.: Curvature measures of subanalytic sets. Amer. J. Math. 116 (1994), no. 4, 819–880.

    Article  MathSciNet  MATH  Google Scholar 

  33. Fu, J.H.G.: Structure of the unitary valuation algebra. J. Differential Geom. 72 (2006), no. 3, 509–533.

    MathSciNet  MATH  Google Scholar 

  34. Gelfand, I. M., Graev, M. I. and Roşu, R.: The problem of integral geometry and intertwining operators for a pair of real Grassmannian manifolds. J. Operator Theory 12 (1984), no. 2, 359–383.

    MathSciNet  Google Scholar 

  35. Gelfand, I. M., Graev, M. I. and Šapiro, Z.Ja.: Integral geometry on k-dimensional planes. (Russian) Funkcional. Anal. i Prilŏzen 1 (1967), 15–31.

    Google Scholar 

  36. Griffiths, P. and Harris, J.: Principles of Algebraic Geometry. Reprint of the 1978 original. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994.

    Google Scholar 

  37. Guillemin, V. and Sternberg, S.: Geometric Asymptotics. Mathematical Surveys, no. 14. American Mathematical Society, Providence, R.I., 1977.

    Google Scholar 

  38. Hadwiger, H.: Translationsinvariante, additive und stetige Eibereichfunktionale. (German) Publ. Math. Debrecen 2 (1951), 81–94.

    Google Scholar 

  39. Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. (German) Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957.

    Google Scholar 

  40. Helgason, S.: The Radon Transform. Second edition. Progress in Math. 5. Birkhäuser Boston, Inc., Boston, MA, 1999.

    Google Scholar 

  41. Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Reprint of the second (1990) edition. Classics Math. Springer-Verlag, Berlin, 2003.

    Google Scholar 

  42. Kashiwara, M. and Schapira, P.: Sheaves on Manifolds. Grundlehren Math. Wiss. 292. Springer-Verlag, Berlin, 1990.

    Google Scholar 

  43. Klain, D.A.: A short proof of Hadwiger’s characterization theorem. Mathematika 42 (1995), no. 2, 329–339.

    Article  MathSciNet  MATH  Google Scholar 

  44. Klain, D.A.: Even valuations on convex bodies. Trans. Amer. Math. Soc. 352 (2000), no. 1, 71–93.

    Article  MathSciNet  MATH  Google Scholar 

  45. Klain, D.A. and Rota, G.-C.: Introduction to Geometric Probability. Lezioni Lincee. Cambridge University Press, Cambridge, 1997.

    Google Scholar 

  46. Khovanskii, A.G. and Pukhlikov, A.V.: Finitely additive measures of virtual polyhedra. (Russian) Algebra i Analiz 4 (1992), no. 2, 161–185; translation in St. Petersburg Math. J. 4 (1993), no. 2, 337–356.

    Google Scholar 

  47. Ludwig, M.: Projection bodies and valuations. Adv. Math. 172 (2002), no. 2, 158–168.

    Article  MathSciNet  MATH  Google Scholar 

  48. Ludwig, M.: Ellipsoids and matrix-valued valuations. Duke Math. J. 119 (2003), no. 1, 159–188.

    Article  MathSciNet  MATH  Google Scholar 

  49. Ludwig, M.: Intersection bodies and valuations. Amer. J. Math. 128 (2006), no. 6, 1409–1428.

    Article  MathSciNet  MATH  Google Scholar 

  50. Ludwig, M. and Reitzner, M.: A characterization of affine surface area. Adv. Math. 147 (1999), no. 1, 138–172.

    Article  MathSciNet  MATH  Google Scholar 

  51. Ludwig, M. and Reitzner, M.: A classification of SL(n) invariant valuations. Ann. of Math. 172 (2010), no. 2, 1219–1267.

    Article  MathSciNet  MATH  Google Scholar 

  52. McMullen, P.: Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. 35 (1977), no. 1, 113–135.

    Article  MathSciNet  MATH  Google Scholar 

  53. McMullen, P.: Continuous translation-invariant valuations on the space of compact convex sets. Arch. Math. (Basel) 34 (1980), no. 4, 377–384.

    Article  MathSciNet  MATH  Google Scholar 

  54. McMullen, P.: Valuations and dissections. In: Handbook of Convex Geometry, Vol. B, 933–988, North-Holland, Amsterdam, 1993.

    Google Scholar 

  55. McMullen, P. and Schneider, R.: Valuations on convex bodies. Convexity and its applications, 170–247, Birkhäuser, Basel, 1983.

    Google Scholar 

  56. SantalĂł, L. A.: Integral Geometry and Geometric Probability. Encyclopedia Math. Appl., 1. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976.

    Google Scholar 

  57. Schapira, P.: Tomography of constructible functions. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (Paris, 1995), 427–435, Lecture Notes in Comput. Sci. 948, Springer, Berlin, 1995.

    Google Scholar 

  58. Schneider, R.: Convex Bodies: the Brunn–Minkowski Theory. Encyclopedia Math. Appl., 44. Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  59. Schneider, R.: Simple valuations on convex bodies. Mathematika 43 (1996), no. 1, 32–39.

    Article  MathSciNet  MATH  Google Scholar 

  60. Schneider, R. and Schuster, F.E.: Rotation equivariant Minkowski valuations. Int. Math. Res. Not. 2006, Art. ID 72894, 20 pp.

    Google Scholar 

  61. Viro, O.: Some integral calculus based on Euler characteristic. In: Topology and Geometry – Rohlin Seminar, 127–138, Lecture Notes in Math. 1346, Springer, Berlin, 1988.

    Google Scholar 

  62. Wintgen, P.: Normal cycle and integral curvature for polyhedra in Riemannian Manifolds. In: Differential Geometry (G. Soos and J. Szenthe, eds.), North-Holland, Amsterdam, 1982.

    Google Scholar 

  63. Zähle, M.: Approximation and characterization of generalised Lipschitz–Killing curvatures. Ann. Global Anal. Geom. 8 (1990), no. 3, 249–260.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Basel

About this chapter

Cite this chapter

Alesker, S. (2014). New Structures on Valuations and Applications. In: Gallego, E., Solanes, G. (eds) Integral Geometry and Valuations. Advanced Courses in Mathematics - CRM Barcelona. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0874-3_1

Download citation

Publish with us

Policies and ethics