Hermitian integral geometry

Abstract

We give in explicit form 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 introduced, essentially, by H. Tasaki. We 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.

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    @article {ale03b, MRKEY = {2015260},
      AUTHOR = {Alesker, Semyon},
      TITLE = {Hard {L}efschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {63},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {63--95},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {52B45 (52A39)},
      MRNUMBER = {2015260},
      MRREVIEWER = {E. Hertel},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1080835658},
      ZBLNUMBER = {1073.52004},
      }
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    @article {ale04, MRKEY = {2053598},
      AUTHOR = {Alesker, Semyon},
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      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {14},
      YEAR = {2004},
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      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {52B45 (13A99)},
      MRNUMBER = {2053598},
      MRREVIEWER = {Jean-Louis Cathelineau},
      DOI = {10.1007/s00039-004-0450-2},
      ZBLNUMBER = {1072.52011},
      }
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    @incollection {ale04a, MRKEY = {2087146},
      AUTHOR = {Alesker, Semyon},
      TITLE = {Hard {L}efschetz theorem for valuations and related questions of integral geometry},
      BOOKTITLE = {Geometric Aspects of Functional Analysis},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1850},
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      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2004},
      MRCLASS = {52B45 (52A22 52A39 53C65)},
      MRNUMBER = {2087146},
      MRREVIEWER = {E. Hertel},
      ZBLNUMBER = {1070.52007},
      }
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    @article {ale-be, MRKEY = {2054020},
      AUTHOR = {Alesker, Semyon and Bernstein, Joseph},
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      AUTHOR = {Alesker, Semyon},
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      YEAR = {2007},
      MRCLASS = {52B45 (32B20 52A39)},
      MRNUMBER = {2347038},
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      }
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      AUTHOR = {Alesker, Semyon},
      TITLE = {Theory of valuations on manifolds: a survey},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {17},
      YEAR = {2007},
      NUMBER = {4},
      PAGES = {1321--1341},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {52B45 (57R99)},
      MRNUMBER = {2373020},
      DOI = {10.1007/s00039-007-0631-x},
      ZBLNUMBER = {1132.52018},
      }
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      }
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      author={Alesker, Semyon},
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      EDITION = {{S}econd},
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      AUTHOR = {Tasaki, Hiroyuki},
      TITLE = {Generalization of {K}ähler angle and integral geometry in complex projective spaces},
      BOOKTITLE = {Steps in Differential Geometry},
      VENUE={{D}ebrecen, 2000},
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      ZBLNUMBER = {0262.53024},
      }

Authors

Andreas Bernig

Goethe-Universität Frankfurt
Frankfurt
Germany

Joseph H. G. Fu

University of Georgia
Athens, GA