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

Monika Ludwig; Matthias Reitzner

doi:http://doi.org/10.4007/annals.2010.172.1219

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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@article {Ludwig:Reitzner, MRKEY = {1725817}, AUTHOR = {Ludwig, Monika and Reitzner, Matthias}, TITLE = {A characterization of affine surface area}, JOURNAL = {Adv. Math.}, FJOURNAL = {Advances in Mathematics}, VOLUME = {147}, YEAR = {1999}, NUMBER = {1}, PAGES = {138--172}, ISSN = {0001-8708}, CODEN = {ADMTA4}, MRCLASS = {52B45 (51M25 52A20 52A38)}, MRNUMBER = {2000j:52018}, MRREVIEWER = {E. Hertel}, DOI = {10.1006/aima.1999.1832}, ZBLNUMBER = {0947.52003}, }
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@article {Lutwak91, MRKEY = {1087796}, AUTHOR = {Lutwak, Erwin}, TITLE = {Extended affine surface area}, JOURNAL = {Adv. Math.}, FJOURNAL = {Advances in Mathematics}, VOLUME = {85}, YEAR = {1991}, NUMBER = {1}, PAGES = {39--68}, ISSN = {0001-8708}, CODEN = {ADMTA4}, MRCLASS = {52A38}, MRNUMBER = {92d:52012}, MRREVIEWER = {G. D. Chakerian}, DOI = {10.1016/0001-8708(91)90049-D}, ZBLNUMBER = {0727.53016}, }
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A classification of $\operatorname{SL}(n)$ invariant valuations

Abstract

Authors