A classification of $\operatorname{SL}(n)$ invariant valuations

Abstract

A classification of upper semicontinuous and $\operatorname{SL}(n)$ invariant valuations on the space of $n$-dimensional convex bodies is established. As a consequence, complete characterizations of centro-affine and $L_p$ affine surface areas are obtained. The proofs make use of a new $\operatorname{SL}(n)$ shaping process for convex bodies.

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      ZBLNUMBER = {0899.52005},
      }
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    @article {Ludwig:affinelength, MRKEY = {1722923},
      AUTHOR = {Ludwig, Monika},
      TITLE = {A characterization of affine length and asymptotic approximation of convex discs},
      JOURNAL = {Abh. Math. Sem. Univ. Hamburg},
      FJOURNAL = {Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg},
      VOLUME = {69},
      YEAR = {1999},
      PAGES = {75--88},
      ISSN = {0025-5858},
      CODEN = {AMHAAJ},
      MRCLASS = {52A10 (52A27)},
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      ZBLNUMBER = {0954.52002},
      }
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    @article {Ludwig:curvature, MRKEY = {1840557},
      AUTHOR = {Ludwig, Monika},
      TITLE = {On the semicontinuity of curvature integrals},
      JOURNAL = {Math. Nachr.},
      FJOURNAL = {Mathematische Nachrichten},
      VOLUME = {227},
      YEAR = {2001},
      PAGES = {99--108},
      ISSN = {0025-584X},
      MRCLASS = {52A22 (52A20)},
      MRNUMBER = {2002c:52007},
      DOI = {10.1002/1522-2616(200107)227:1<99::AID-MANA99>3.3.CO;2-H},
      ZBLNUMBER = {0998.52003},
      }
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    @article {Ludwig:projection, MRKEY = {1942402},
      AUTHOR = {Ludwig, Monika},
      TITLE = {Projection bodies and valuations},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {172},
      YEAR = {2002},
      NUMBER = {2},
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      CODEN = {ADMTA4},
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      MRREVIEWER = {J. M. Wills},
      DOI = {10.1016/S0001-8708(02)00021-X},
      ZBLNUMBER = {1019.52003},
      }
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    @article {Ludwig:origin, MRKEY = {1932331},
      AUTHOR = {Ludwig, Monika},
      TITLE = {Valuations of polytopes containing the origin in their interiors},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {170},
      YEAR = {2002},
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      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {52B45 (52A38)},
      MRNUMBER = {2003g:52022},
      MRREVIEWER = {Uwe Schnell},
      URL = {http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=MR&_method=citationSearch&_piikey=S0001870802920773&_version=1&md5=b814cfdb5f3ca12e4c47774399dccb85},
      ZBLNUMBER = {1015.52012},
      }
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    @article {Ludwig:matrix, MRKEY = {1991649},
      AUTHOR = {Ludwig, Monika},
      TITLE = {Ellipsoids and matrix-valued valuations},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {119},
      YEAR = {2003},
      NUMBER = {1},
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      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {52B45 (52B11)},
      MRNUMBER = {2004e:52015},
      MRREVIEWER = {E. Hertel},
      DOI = {10.1215/S0012-7094-03-11915-8},
      ZBLNUMBER = {1033.52012},
      }
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    @article {Ludwig:intersection, MRKEY = {2275906},
      AUTHOR = {Ludwig, Monika},
      TITLE = {Intersection bodies and valuations},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {128},
      YEAR = {2006},
      NUMBER = {6},
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      CODEN = {AJMAAN},
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      MRREVIEWER = {Peter M. Gruber},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v128/128.6ludwig.pdf},
      ZBLNUMBER = {1115.52007},
      }
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    @article {Ludwig:Reitzner, MRKEY = {1725817},
      AUTHOR = {Ludwig, Monika and Reitzner, Matthias},
      TITLE = {A characterization of affine surface area},
      JOURNAL = {Adv. Math.},
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      VOLUME = {147},
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      CODEN = {ADMTA4},
      MRCLASS = {52B45 (51M25 52A20 52A38)},
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      }
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    @article {Ludwig:Reitzner_moves, MRKEY = {2225672},
      AUTHOR = {Ludwig, Monika and Reitzner, Matthias},
      TITLE = {Elementary moves on triangulations},
      JOURNAL = {Discrete Comput. Geom.},
      FJOURNAL = {Discrete \& Computational Geometry. An International Journal of Mathematics and Computer Science},
      VOLUME = {35},
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      AUTHOR = {Lutwak, Erwin},
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      MRREVIEWER = {G. D. Chakerian},
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      AUTHOR = {Lutwak, Erwin},
      TITLE = {The {B}runn-{M}inkowski-{F}irey theory. {I}. {M}ixed volumes and the {M}inkowski problem},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {38},
      YEAR = {1993},
      NUMBER = {1},
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      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {52A39},
      MRNUMBER = {94g:52008},
      MRREVIEWER = {P. R. Goodey},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214454097},
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      }
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      AUTHOR = {Lutwak, Erwin},
      TITLE = {The {B}runn-{M}inkowski-{F}irey theory. {II}. {A}ffine and geominimal surface areas},
      JOURNAL = {Adv. Math.},
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      AUTHOR = {Lutwak, Erwin and Oliker, Vladimir},
      TITLE = {On the regularity of solutions to a generalization of the {M}inkowski problem},
      JOURNAL = {J. Differential Geom.},
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      VOLUME = {41},
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      AUTHOR = {Lutwak, Erwin and Yang, Deane and Zhang, Gaoyong},
      TITLE = {A new ellipsoid associated with convex bodies},
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      AUTHOR = {Lutwak, Erwin and Yang, Deane and Zhang, Gaoyong},
      TITLE = {{$L\sb p$} affine isoperimetric inequalities},
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      AUTHOR = {Lutwak, Erwin and Yang, Deane and Zhang, Gaoyong},
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      AUTHOR = {Lutwak, Erwin and Yang, Deane and Zhang, Gaoyong},
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      JOURNAL = {Duke Math. J.},
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      AUTHOR = {Lutwak, Erwin and Yang, Deane and Zhang, Gaoyong},
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      JOURNAL = {J. Differential Geom.},
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      TITLE = {Valuations and dissections},
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      JOURNAL = {Int. Math. Res. Not.},
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Authors

Monika Ludwig

Institut für Diskrete Mathematik and Geometrie
TU Wien
Wiedner Hauptstrasse 8-10/104
1040 Wien
Austria
and
Department of Mathematics
Polytechnic Institute of New York University
Six Metrotech Center
Brooklyn, New York 11201
United States

Matthias Reitzner

Fachbereich 6: Mathematik/Informatik
Universität Osnarbück
Albrechtstrasse 28a
49076 Osnabrück
Germany