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Hermitian integral geometry. (English) Zbl 1230.52014

The authors give in explicit form of the principal kinematic formula for the action of the affine unitary group on \(\mathbb{C}^n\), together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations. The authors introduce also several other canonical bases for the algebra of unitary-invariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone elements.

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
53C65 Integral geometry
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References:

[1] S. Alesker, ”Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations,” J. Differential Geom., vol. 63, iss. 1, pp. 63-95, 2003. · Zbl 1073.52004
[2] S. Alesker, ”The multiplicative structure on continuous polynomial valuations,” Geom. Funct. Anal., vol. 14, iss. 1, pp. 1-26, 2004. · Zbl 1072.52011 · doi:10.1007/s00039-004-0450-2
[3] S. Alesker, ”Hard Lefschetz theorem for valuations and related questions of integral geometry,” in Geometric Aspects of Functional Analysis, New York: Springer-Verlag, 2004, vol. 1850, pp. 9-20. · Zbl 1070.52007
[4] S. Alesker and J. Bernstein, ”Range characterization of the cosine transform on higher Grassmannians,” Adv. Math., vol. 184, iss. 2, pp. 367-379, 2004. · Zbl 1059.22013 · doi:10.1016/S0001-8708(03)00149-X
[5] S. Alesker, ”Theory of valuations on manifolds. I. Linear spaces,” Israel J. Math., vol. 156, pp. 311-339, 2006. · Zbl 1132.52017 · doi:10.1007/BF02773837
[6] S. Alesker, ”Theory of valuations on manifolds. II,” Adv. Math., vol. 207, iss. 1, pp. 420-454, 2006. · Zbl 1117.52016 · doi:10.1016/j.aim.2005.11.015
[7] S. Alesker, ”Theory of valuations on manifolds. IV. New properties of the multiplicative structure,” in Geometric Aspects of Functional Analysis, New York: Springer-Verlag, 2007, vol. 1910, pp. 1-44. · Zbl 1127.52010 · doi:10.5565/PUBLMAT_50106_12
[8] S. Alesker, ”Theory of valuations on manifolds: a survey,” Geom. Funct. Anal., vol. 17, iss. 4, pp. 1321-1341, 2007. · Zbl 1132.52018 · doi:10.1007/s00039-007-0631-x
[9] S. Alesker and J. H. G. Fu, ”Theory of valuations on manifolds. III. Multiplicative structure in the general case,” Trans. Amer. Math. Soc., vol. 360, iss. 4, pp. 1951-1981, 2008. · Zbl 1130.52008 · doi:10.1090/S0002-9947-07-04489-3
[10] S. Alesker, A Fourier type transform on translation invariant valuations on convex sets, preprint, 2011. · Zbl 1219.52009 · doi:10.1007/s11856-011-0008-6
[11] A. Bernig, ”A Hadwiger-type theorem for the special unitary group,” Geom. Funct. Anal., vol. 19, iss. 2, pp. 356-372, 2009. · Zbl 1180.53076 · doi:10.1007/s00039-009-0008-4
[12] A. Bernig and L. Bröcker, ”Valuations on manifolds and Rumin cohomology,” J. Differential Geom., vol. 75, iss. 3, pp. 433-457, 2007. · Zbl 1117.58005
[13] A. Bernig and J. H. G. Fu, ”Convolution of convex valuations,” Geom. Dedicata, vol. 123, pp. 153-169, 2006. · Zbl 1117.53054 · doi:10.1007/s10711-006-9115-7
[14] . Chern, ”On the kinematic formula in integral geometry,” J. Math. Mech., vol. 16, pp. 101-118, 1966. · Zbl 0142.20704
[15] P. Deligne, ”La conjecture de Weil. II,” Inst. Hautes Études Sci. Publ. Math., iss. 52, pp. 137-252, 1980. · Zbl 0456.14014 · doi:10.1007/BF02684780
[16] H. Federer, ”Curvature measures,” Trans. Amer. Math. Soc., vol. 93, pp. 418-491, 1959. · Zbl 0089.38402 · doi:10.2307/1993504
[17] J. H. G. Fu, ”Kinematic formulas in integral geometry,” Indiana Univ. Math. J., vol. 39, iss. 4, pp. 1115-1154, 1990. · Zbl 0703.53059 · doi:10.1512/iumj.1990.39.39052
[18] J. H. G. Fu, ”Curvature measures of subanalytic sets,” Amer. J. Math., vol. 116, iss. 4, pp. 819-880, 1994. · Zbl 0818.53091 · doi:10.2307/2375003
[19] J. H. G. Fu, ”Structure of the unitary valuation algebra,” J. Differential Geom., vol. 72, iss. 3, pp. 509-533, 2006. · Zbl 1096.52003
[20] J. H. G. Fu, ”Integral geometry and Alesker’s theory of valuations,” in Integral Geometry and Convexity, World Sci. Publ., Hackensack, NJ, 2006, pp. 17-27. · Zbl 1121.52029 · doi:10.1142/9789812774644_0002
[21] P. Griffiths and J. Harris, Principles of Algebraic Geometry, New York: Wiley-Interscience [John Wiley & Sons], 1978. · Zbl 0408.14001
[22] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, New York: Springer-Verlag, 1957, vol. 93. · Zbl 0078.35703
[23] R. Howard, ”The kinematic formula in Riemannian homogeneous spaces,” Mem. Amer. Math. Soc., vol. 106, iss. 509, p. vi, 1993. · Zbl 0810.53057
[24] D. Huybrechts, Complex Geometry, New York: Springer-Verlag, 2005. · Zbl 1055.14001 · doi:10.1007/b137952
[25] H. J. Kang and H. Tasaki, ”Integral geometry of real surfaces in the complex projective plane,” Geom. Dedicata, vol. 90, pp. 99-106, 2002. · Zbl 1008.53056 · doi:10.1023/A:1014933627990
[26] D. A. Klain, ”Even valuations on convex bodies,” Trans. Amer. Math. Soc., vol. 352, iss. 1, pp. 71-93, 2000. · Zbl 0940.52002 · doi:10.1090/S0002-9947-99-02240-0
[27] D. A. Klain and G. Rota, Introduction to Geometric Probability, Cambridge: Cambridge Univ. Press, 1997. · Zbl 0896.60004
[28] A. Nijenhuis, ”On Chern’s kinematic formula in integral geometry,” J. Differential Geometry, vol. 9, pp. 475-482, 1974. · Zbl 0284.53045
[29] H. Park, Kinematic formulas for the real subspaces of complex space forms of dimension \(2\) and \(3\), Ph.D. thesis, University of Georgia, 2002.
[30] M. Rumin, ”Formes différentielles sur les variétés de contact,” J. Differential Geom., vol. 39, iss. 2, pp. 281-330, 1994. · Zbl 0973.53524
[31] L. A. Santaló, Integral Geometry and Geometric Probability, Second ed., Cambridge: Cambridge Univ. Press, 2004. · Zbl 1116.53050
[32] R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Cambridge: Cambridge Univ. Press, 1993, vol. 44. · Zbl 0798.52001 · doi:10.1017/CBO9780511526282
[33] H. Tasaki, ”Generalization of Kähler angle and integral geometry in complex projective spaces,” in Steps in Differential Geometry, Inst. Math. Inform., Debrecen, 2001, pp. 349-361. · Zbl 0984.53030
[34] H. Tasaki, ”Generalization of Kähler angle and integral geometry in complex projective spaces. II,” Math. Nachr., vol. 252, pp. 106-112, 2003. · Zbl 1038.53069 · doi:10.1002/mana.200310040
[35] K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry, New York: Marcel Dekker, 1973, vol. 16. · Zbl 0262.53024
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