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Characterisation of valuations and curvature measures in euclidean spaces

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Abstract

Valuations constitute a class of functionals on convex bodies which include the Euler-characteristic, the surface area, the Lebesgue-measure, and many more classical functionals. Curvature measures may be regarded as “localised” versions of valuations which yield local information about the geometry of a body’s boundary.

A complete classification of continuous translation-invariant SO(n)-invariant valuations and curvature measures with values in ℝ was obtained by Hadwiger and Schneider, respectively. More recently, characterisation results have been achieved for curvature measures with values in Symp ℝn and Sym2Λqℝn for p, q ≥ 1 with varying assumptions as for their invariance properties.

In the present work, we classify all smooth translation-invariant SO(n)-equivariant curvature measures with values in any SO(n)-representation in terms of certain differential forms on the sphere bundle Sℝn and describe their behaviour under the globalisation map. The latter result also yields a similar classification of all continuous SO(n)-equivariant valuations with values in any SO(n)-representation. Furthermore, a decomposition of the space of smooth translation-invariant ℝ-valued curvature measures as an SO(n)-representation is obtained. As a corollary, we construct an explicit basis of continuous translation-invariant ℝ-valued valuations.

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References

  1. J. Abardia, Difference bodies in complex vector spaces, Journal of Functional Analysis 263 (2012), 3588–3603.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Alesker, Description of continuous isometry equivariant valuations on convex sets, Geometriae Dedicata 74 (1999), 241–248.

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture, Geometric and Functional Analysis 11 (2001), 244–272.

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Alesker, Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations, Journal of Differential Geometry 63 (2003), 63–95.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. Alesker, The multiplicative structure on continuous polynomial valuations, Geometric and Functional Analysis 14 (2004), 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Alesker, Theory of valuations on manifolds I. Linear spaces., Israel Journal of Mathematics 156 (2006), 311–339.

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Alesker, Theory of valuations on manifolds II, Advances in Mathematics 207 (2006), 420–454.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Alesker, Plurisubharmonic functions on the octonionic plane and Spin(9)-invariant valuations on convex sets, Journal of Geometric Analysis 18 (2008), 651–686.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Alesker, A Fourier type transform on translation invariant valuations on convex sets, Israel Journal of Mathematics 181 (2011), 189–294.

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Alesker, Kotrbaty’s theorem on valuations and geometric inequalities for convex bodies, Israel Journal of Mathematics 247 (2022), 361–378.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Alesker and A. Bernig, The product on smooth and generalized valuations, American Journal of Mathematics 134 (2012), 507–560.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Alesker, A. Bernig and F. Schuster, Harmonic analysis of translation invariant valuations, Geometric and Functional Analysis 21 (2011), 751–773.

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Alesker and J. H. G. Fu, Theory of valuations on manifolds III. Multiplicative structure in the general case, Transactions of the American Mathematical Society 360 (2008), 1951–1981.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Bernig, Curvature tensors of singular spaces, Differential Geometry and its Applications 24 (2005), 191–208.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Bernig, A Hadwiger-type theorem for the special unitary group, Geometric and Functional Analysis 19 (2009), 356–372.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Bernig, Integral geometry under G2and Spin(7), Israel Journal of Mathematics 184 (2011), 301–316.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Bernig, Algebraic integral geometry, in Global Differential Geometry, Springer Proceedings in Mathematics, Vol. 17, Springer, Berlin-Heidelberg, 2012, pp. 107–146.

    Chapter  MATH  Google Scholar 

  18. A. Bernig and L. Bröcker, Valuations on manifolds and Rumin cohomology, Journal of Differential Geometry 75 (2007), 433–457.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Bernig and D. Faifman, Valuation theory of indefinite orthogonal groups, Journal of Functional Analysis 273 (2017), 2167–2247.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Bernig and J. H. G. Fu, Hermitian integral geometry, Annals of Mathematics 173 (2011), 907–945.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Bernig and D. Hug, Kinematic formulas for tensor valuations, Journal für die reine und angewandte Mathematik 736 (2018), 141–191.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Bernig and G. Solanes, Classification of invariant valuations on the quaternionic plane, Journal of Functional Analysis 267 (2014), 2933–2961.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Bernig and G. Solanes, Kinematic formulas on the quaternionic plane, Proceedings of the London Mathematical Society 115 (2017), 725–762.

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Dorrek and F. E. Schuster, Projection functions, area measures and the Alesker-Fourier transform, Journal of Functional Analysis 273 (2017), 2026–2069.

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Federer, Curvature measures, Transactions of the American Mathematical Society 93 (1959), 418–491.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. H. G. Fu, Kinematic formulas in integral geometry, Indiana University Mathematics Journal 39 (1990), 1115–1154.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. H. G. Fu, Algebraic integral geometry, in Integral Geometry and Valuations, Advanced Courses in Mathematics. CRM Barcelona, Bithäuser/Springer, Basel, 2014, pp. 47–112.

    Google Scholar 

  29. J. H. G. Fu, D. Pokorny and J. Rataj, Kinematic formulas for sets defined by differences of convex functions, Advances in Mathematics 311 (2017), 796–832.

    Article  MathSciNet  MATH  Google Scholar 

  30. W. Fulton, Young Tableaux, London mathematical Society Student Texts, Vol. 35, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  31. W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, Vol. 129, Springer, New York, 1999.

    MATH  Google Scholar 

  32. R. Goodman and N. R. Wallach, Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and its Applications, Vol. 68, Cambridge University Press, Cambridge, 2003.

    MATH  Google Scholar 

  33. H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin—Güottingen—Heidelberg, 1957.

    Book  MATH  Google Scholar 

  34. H. Hadwiger and R. Schneider, Vektorielle Integralgeometrie, Elemente der Mathematik 26 (1971), 49–57.

    MathSciNet  MATH  Google Scholar 

  35. D. Hug and R. Schneider, Local tensor valuations, Geometric and Functional Analysis 24 (2014), 1516–1564.

    Article  MathSciNet  MATH  Google Scholar 

  36. D. Hug and R. Schneider, Rotation equivariant local tensor valuations on convex bodies, Communications in Contemporary Mathematics 19 (2017), Article no. 1650061.

  37. D. Hug and R. Schneider, SO(n) equivariant local tensor valuations on polytopes, Michigan Mathematical Journal 66 (2017), 637–659.

    Article  MathSciNet  MATH  Google Scholar 

  38. D. Hug, R. Schneider and R. Schuster, The space of isometry equivariant tensor valuations, Algebra i Analiz 19 (2007), 194–224.

    Google Scholar 

  39. D. Hug, R. Schneider and R. Schuster, Integral geometry of tensor valuations, Advances in Applied Mathematics 41 (2008), 482–509.

    Article  MathSciNet  MATH  Google Scholar 

  40. D. Hug and J. A. Weis, Kinematic formulae for tensorial curvature measures, Annali di Matematica Pura ed Applicata 197 (2018), 1349–1384.

    Article  MathSciNet  MATH  Google Scholar 

  41. D. Hug and J. A. Weis, Integral geometric formulae for Minkowski tensors, https://arxiv.org/abs/1712.09699.

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

    MATH  Google Scholar 

  43. A. W. Knapp, Representation Theory of Semisimple Groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 2001.

    MATH  Google Scholar 

  44. J. Kotrbatý, On Hodge-Riemann relations for translation-invariant valuations, Advances in Mathematics 390 (2021), Article no. 107914.

  45. J. Kotrbatý and T. Wannerer, On mixed Hodge-Riemann relations for translation-invariant valuations and Aleksandrov-Fenchel inequalities, Communications in Contemporary Mathematics, in press, https://doi.org/10.1142/S0219199721500498.

  46. M. Ludwig, Minkowski valuations, Transactions of the American Mathematical Society 357 (2005), 4191–4213.

    Article  MathSciNet  MATH  Google Scholar 

  47. M. Ludwig and M. Reitzner, A characterization of affine surface area, Advances in Mathematics 147 (1999), 138–172.

    Article  MathSciNet  MATH  Google Scholar 

  48. M. Ludwig and M. Reitzner, A classification of SL(n) invariant valuations, Annals of Mathematics 172 (2010), 1219–1267.

    Article  MathSciNet  MATH  Google Scholar 

  49. I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  51. P. McMullen, Isometry equivariant valuations on convex bodies, Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento 50 (1997), 259–271.

    MATH  Google Scholar 

  52. R. Schneider, Curvature measures of convex bodies, Annali di Matematica Pura ed Applicata 116 (1978), 101–134.

    Article  MathSciNet  MATH  Google Scholar 

  53. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, Vol. 151, Cambridge University Press, Cambridge, 2014.

    MATH  Google Scholar 

  54. F. Schuster and T. Wannerer, GL(n) contravariant Minkowski valuations, Transactions of the American Mathematical Society 364 (2012), 815–826.

    Article  MathSciNet  MATH  Google Scholar 

  55. M. R. Sepanski, Compact Lie Groups, Graduate Texts in Mathematics, Vol. 235, Springer, New York, 2007.

    MATH  Google Scholar 

  56. J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer, New York-Heidelberg, 1977.

    MATH  Google Scholar 

  57. G. Solanes, Contact measures in isotropic spaces, Advances in Mathematics 317 (2017), 645–664.

    Article  MathSciNet  MATH  Google Scholar 

  58. T. Wannerer, GL(n) equivariant Minkowski valuations, Indiana University Mathematics Journal 60 (2011), 1655–1672.

    Article  MathSciNet  MATH  Google Scholar 

  59. T. Wannerer, Integral geometry of unitary area measures, Advances in Mathematics 263 (2014), 1–44.

    Article  MathSciNet  MATH  Google Scholar 

  60. T. Wannerer, The module of unitarily invariant area measures, Journal of Differential Geometry 96 (2014), 1895–1922.

    Article  MathSciNet  MATH  Google Scholar 

  61. M. Zähle, Integral and current representation of Federer’s curvature measures, Archiv der Mathematik 46 (1986), 557–567.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60054, Frankfurt, Germany

    Mykhailo Saienko

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Correspondence to Mykhailo Saienko.

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Supported by DFG grants BE 2484/5-1 and BE 2484/5-2.

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Saienko, M. Characterisation of valuations and curvature measures in euclidean spaces. Isr. J. Math. 250, 463–500 (2022). https://doi.org/10.1007/s11856-022-2343-1

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  • DOI: https://doi.org/10.1007/s11856-022-2343-1

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