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Transactions of the American Mathematical Society

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SL($ n$) covariant vector valuations on polytopes


Authors: Chunna Zeng and Dan Ma
Journal: Trans. Amer. Math. Soc. 370 (2018), 8999-9023
MSC (2010): Primary 52B45, 52A20
DOI: https://doi.org/10.1090/tran/7468
Published electronically: August 17, 2018
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Abstract: All $ \mathrm {SL}(n)$ covariant vector valuations on convex polytopes in $ \mathbb{R}^n$ are completely classified without any continuity assumptions. The moment vector turns out to be the only such valuation if $ n\ge 3$, while two new functionals show up in dimension two.


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Chunna Zeng
Affiliation: School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, People’s Republic of China; and Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8–10/1046, 1040 Wien, Austria
Email: zengchn@163.com

Dan Ma
Affiliation: Department of Mathematics, Shanghai Normal University, Shanghai 200234, People’s Republic of China
Email: madan@shnu.edu.cn

DOI: https://doi.org/10.1090/tran/7468
Keywords: Moment vector, valuation, convex polytope, $\mathrm{SL}(n)$ covariance
Received by editor(s): April 25, 2017
Received by editor(s) in revised form: November 20, 2017
Published electronically: August 17, 2018
Additional Notes: The first author was supported in part by the National Natural Science Foundation of China (Project No. 11801048), by the Chinese Scholarship Council and by the Natural Science Foundation Project of CSTC (Grant No. cstc2017jcyjAX0022).
The second author was supported in part by Shanghai Sailing Program 17YF1413800 and by the National Natural Science Foundation of China (Project No. 11701373). The second author is the corresponding author.
Article copyright: © Copyright 2018 American Mathematical Society

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