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Transactions of the American Mathematical Society

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$ L_p$-Blaschke valuations


Authors: Jin Li, Shufeng Yuan and Gangsong Leng
Journal: Trans. Amer. Math. Soc. 367 (2015), 3161-3187
MSC (2010): Primary 52B45, 52A20
DOI: https://doi.org/10.1090/S0002-9947-2015-06047-4
Published electronically: January 20, 2015
MathSciNet review: 3314805
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Abstract: In this article, a classification of continuous, linearly intertwining, symmetric $ L_p$-Blaschke ($ p>1$) valuations is established as an extension of Haberl's work on Blaschke valuations. More precisely, we show that for dimensions $ n \geq 3$, the only continuous, linearly intertwining, normalized symmetric $ L_p$-Blaschke valuation is the normalized $ L_p$-curvature image operator, while for dimension $ n = 2 $, a rotated normalized $ L_p$-curvature image operator is the only additional one. One of the advantages of our approach is that we deal with normalized symmetric $ L_p$-Blaschke valuations, which makes it possible to handle the case $ p=n$. The cases where $ p \not =n$ are also discussed by studying the relations between symmetric $ L_p$-Blaschke valuations and normalized ones.


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Additional Information

Jin Li
Affiliation: Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China
Email: lijin2955@gmail.com

Shufeng Yuan
Affiliation: Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China
Email: yuanshufeng2003@163.com

Gangsong Leng
Affiliation: Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China
Email: gleng@staff.shu.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-2015-06047-4
Keywords: Normalized $L_p$-Blaschke valuation, normalized $L_p$-curvature image, $L_p$-Blaschke valuation, $L_p$-curvature image
Received by editor(s): September 22, 2012
Received by editor(s) in revised form: November 19, 2012, and December 10, 2012
Published electronically: January 20, 2015
Additional Notes: The authors would like to acknowledge the support from the National Natural Science Foundation of China (11271244), Shanghai Leading Academic Discipline Project (S30104), and Innovation Foundation of Shanghai University (SHUCX120121).
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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