Algebraic Integral Geometry

Conference paper
First Online:
Part of the Springer Proceedings in Mathematics book series (PROM, volume 17)

Abstract

A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.

Keywords

Convex Body Integral Geometry Intrinsic Volume Euclidean Vector Space Invariant Valuation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Notes

Acknowledgements

I was happy to profit from many discussions with and talks by Semyon Alesker and Joseph Fu on algebraic integral geometry. The present text is strongly influenced by their ideas and I am grateful to them. The terms Algebraic integral geometryand Fundamental theorem of algebraic integral geometrywere invented by Fu. I also thank Gautier Berck, Franz Schuster and Christoph Thäle for numerous useful remarks on this text.

References

  1. 1.
    Abardia, J.: Geometria integral en espais de curvatura holomorfa constant. PhD-Thesis, Universitat Autònoma de Barcelona (2009)Google Scholar
  2. 2.
    Abardia, J., Bernig, A.: Projection bodies in complex vector spaces. Adv. Math. 227(2), 830–846 (2011)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Abardia, J., Gallego, E., Solanes, G.: Gauss-Bonnet theorem and Crofton type formulas in complex space forms. Israel J. Math., arXiv:0904.0336 (to appear)Google Scholar
  4. 4.
    Alesker, S.: Description of continuous isometry covariant valuations on convex sets. Geom. Dedicata 74(3), 241–248 (1999)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Alesker S.: Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture. Geom. Funct. Anal. 11(2), 244–272 (2001)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Alesker, S.: Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations. J. Diff. Geom. 63(1), 63–95 (2003)MATHMathSciNetGoogle Scholar
  7. 7.
    Alesker, S.: Hard Lefschetz theorem for valuations and related questions of integral geometry. In: Geometric aspects of functional analysis, vol. 1850 of Lecture Notes in Math., pp. 9–20. Springer, Berlin (2004)Google Scholar
  8. 8.
    Alesker, S.: SU(2)-invariant valuations. In: Milman, V.D. et al. (eds.) Geometric Aspects of Functional Analysis. Papers from the Israel seminar (GAFA) 2002–2003. Lecture Notes in Mathematics, vol. 1850, pp. 21–29. Springer, Berlin (2004)Google Scholar
  9. 9.
    Alesker, S.: The multiplicative structure on continuous polynomial valuations. Geom. Funct. Anal. 14(1), 1–26 (2004)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Alesker, S.: Theory of valuations on manifolds. I: Linear spaces. Isr. J. Math. 156, 311–339 (2006)MATHMathSciNetGoogle Scholar
  11. 11.
    Alesker, S.: Theory of valuations on manifolds. II. Adv. Math. 207(1), 420–454 (2006)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Alesker, S.: Theory of valuations on manifolds: A survey. Geom. Funct. Anal. 17(4), 1321–1341 (2007)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Alesker, S.: Theory of valuations on manifolds. IV. New properties of the multiplicative structure. In: Geometric aspects of functional analysis, vol. 1910 of Lecture Notes in Math., pp. 1–44. Springer, Berlin (2007)Google Scholar
  14. 14.
    Alesker, S.: A Fourier type transform on translation invariant valuations on convex sets. Israel J. Math. 181, 189–294 (2011)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Alesker, S., Bernig, A.: The product on smooth and generalized valuations. Am. J. Math. (to appear)Google Scholar
  16. 16.
    Alesker, S., Bernstein, J.: Range characterization of the cosine transform on higher Grassmannians. Adv. Math. 184(2), 367–379 (2004)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Alesker, S., Fu, J.H.G.: Theory of valuations on manifolds. III. Multiplicative structure in the general case. Trans. Am. Math. Soc. 360(4), 1951–1981 (2008)MATHMathSciNetGoogle Scholar
  18. 18.
    Bernig, A.: Integral geometry under G 2and Spin(7). Israel J. Math. (to appear)Google Scholar
  19. 19.
    Bernig, A.: Invariant valuations on quaternionic vector spaces. J. Inst. Math. Jussieu (to appear)Google Scholar
  20. 20.
    Bernig, A.: The normal cycle of a compact definable set. Israel J. Math. 159, 373–411 (2007)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Bernig, A.: A Hadwiger type theorem for the special unitary group. Geom. Funct. Anal. 19, 356–372 (2009)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Bernig, A.: A product formula for valuations on manifolds with applications to the integral geometry of the quaternionic line. Comment. Math. Helv. 84(1), 1–19 (2009)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Bernig, A., Bröcker, L.: Valuations on manifolds and Rumin cohomology. J. Differ. Geom. 75(3), 433–457 (2007)MATHGoogle Scholar
  24. 24.
    Bernig, A., Fu, J.H.G.: Convolution of convex valuations. Geom. Dedicata 123, 153–169 (2006)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Bernig, A., Fu, J.H.G.: Hermitian integral geometry. Ann. Math. 173, 907–945 (2011)MATHMathSciNetGoogle Scholar
  26. 26.
    Besse, A.L.: Einstein manifolds. Classics in Mathematics. Springer, Berlin (2008) Reprint of the 1987 editionGoogle Scholar
  27. 27.
    Borel, A.: Some remarks about Lie groups transitive on spheres and tori. Bull. Am. Math. Soc. 55, 580–587 (1949)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Fu, J.H.G.: Algebraic integral geometry. Preprint.Google Scholar
  29. 29.
    Fu, J.H.G.: Kinematic formulas in integral geometry. Indiana Univ. Math. J. 39(4), 1115–1154 (1990)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Fu, J.H.G.: Curvature measures of subanalytic sets. Am. J. Math. 116(4), 819–880 (1994)CrossRefMATHGoogle Scholar
  31. 31.
    Fu, J.H.G.: Integral geometry and Alesker’s theory of valuations. In: Integral geometry and convexity, pp. 17–27. World Sci. Publ., Hackensack, NJ (2006)Google Scholar
  32. 32.
    Fu, J.H.G.: Structure of the unitary valuation algebra. J. Diff. Geom. 72(3), 509–533 (2006)MATHGoogle Scholar
  33. 33.
    Fu, J.H.G.: The two faces of Blaschkean integral geometry. Lecture notes from the Summer school. New approaches to curvature, Les Diablerets September 2008Google Scholar
  34. 34.
    Howard, R.: The kinematic formula in Riemannian homogeneous spaces. Mem. Am. Math. Soc. 106(509), vi+69 (1993)Google Scholar
  35. 35.
    Hug, D., Schneider, R., Schuster, R.: The space of isometry covariant tensor valuations. Algebra i Analiz 19(1), 194–224 (2007)MathSciNetGoogle Scholar
  36. 36.
    Hug, D., Schneider, R., Schuster, R.: Integral geometry of tensor valuations. Adv. Appl. Math. 41(4) 482–509 (2008)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Huybrechts, D.: Complex geometry. Universitext. An introduction. Springer, Berlin, (2005)MATHGoogle Scholar
  38. 38.
    Kazarnovskiĭ, B.Ja.: On zeros of exponential sums. Dokl. Akad. Nauk SSSR 257(4), 804–808 (1981)Google Scholar
  39. 39.
    Klain, D.A.: Even valuations on convex bodies. Trans. Am. Math. Soc. 352(1), 71–93 (2000)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Klain, D.A., Rota, G.-C.: Introduction to geometric probability. Lezioni Lincee. [Lincei Lectures]. Cambridge University Press, Cambridge (1997)Google Scholar
  41. 41.
    Ludwig, M.: Intersection bodies and valuations. Am. J. Math. 128(6), 1409–1428 (2006)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Ludwig, M.: Valuations in the affine geometry of convex bodies. In: Integral Geometry and Convexity, pp. 49–65 (2006)Google Scholar
  43. 43.
    Ludwig, M., Reitznerm M.: A characterization of affine surface area. Adv. Math. 147(1), 138–172 (1999)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Lutwak, E.: Extended affine surface area. Adv. Math. 85(1), 39–68 (1991)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    McMullen, P.: Valuations and Euler-type relations on certain classes of convex polytopes. Proc. London Math. Soc. (3) 35(1), 113–135 (1977)Google Scholar
  46. 46.
    McMullen, P.: Continuous translation-invariant valuations on the space of compact convex sets. Arch. Math. (Basel) 34(4), 377–384 (1980)Google Scholar
  47. 47.
    McMullen, P.: Isometry covariant valuations on convex bodies. Rend. Circ. Mat. Palermo (2) Suppl. (50), 259–271 (1997); II International Conference in “Stochastic Geometry, Convex Bodies and Empirical Measures” (Agrigento, 1996)Google Scholar
  48. 48.
    Montgomery, D., Samelson, H.: Transformation groups of spheres. Ann. Math. (2) 44, 454–470 (1943)Google Scholar
  49. 49.
    Nijenhuis, A.: On Chern’s kinematic formula in integral geometry. J. Diff. Geom. 9, 475–482 (1974)MATHMathSciNetGoogle Scholar
  50. 50.
    Park, H.: Kinematic formulas for the real subspaces of complex space forms of dimension 2 and 3. PhD-thesis University of Georgia 2002Google Scholar
  51. 51.
    Rumin, M.: Differential forms on contact manifolds. (Formes différentielles sur les variétés de contact.). J. Differ. Geom. 39(2), 281–330 (1994)Google Scholar
  52. 52.
    Santaló, L.A.: Integral geometry and geometric probability. In: With a foreword by Mark Kac, Encyclopedia of Mathematics and its Applications, vol. 1. Addison-Wesley, Reading (1976)Google Scholar
  53. 53.
    Schneider, R.: Convex bodies: the Brunn-Minkowski theory, vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1993)Google Scholar
  54. 54.
    Schneider, R.: Simple valuations on convex bodies. Mathematika 43(1), 32–39 (1996)CrossRefMATHMathSciNetGoogle Scholar
  55. 55.
    Schuster, F.E.: Valuations and Busemann-Petty type problems. Adv. Math. 219(1), 344–368 (2008)CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Tasaki, H.: Generalization of Kähler angle and integral geometry in complex projective spaces. In: Steps in differential geometry (Debrecen, 2000), pp. 349–361. Inst. Math. Inform., Debrecen (2001)Google Scholar
  57. 57.
    Tasaki, H.: Generalization of Kähler angle and integral geometry in complex projective spaces. II. Math. Nachr. 252, 106–112 (2003)CrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    Valuations and Integral Geometry. Abstracts from the mini-workshop held January 17–23, 2010. Organized by Semyon Alesker, Andreas Bernig and Franz SchusterGoogle Scholar
  59. 59.
    van den Dries, L.: Tame topology and o-minimal structures, vol. 248 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institut für MathematikGoethe-Universität FrankfurtFrankfurtGermany

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