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LYZ Matrices and SL($n$) Contravariant Valuations on Polytopes

Part of: General convexity Polytopes and polyhedra

Published online by Cambridge University Press:  18 December 2019

Dan Ma  and
Wei Wang
Dan Ma
Affiliation:
Department of Mathematics, Shanghai Normal University, Shanghai, 200234, P.R.China, Email: madan@shnu.edu.cn
Wei Wang
Affiliation:
School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, 411201, P.R. China, Email: wwang@hnust.edu.cn
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Abstract

All SL($n$) contravariant symmetric matrix valued valuations on convex polytopes in $\mathbb{R}^{n}$ are completely classified without any continuity assumptions. The general Lutwak–Yang–Zhang matrix is shown to be essentially the unique such valuation.

Keywords

LYZ matrixvaluationconvex polytopeSL(n) contravariance

MSC classification

Primary: 52B45: Dissections and valuations (Hilbert's third problem, etc.)
Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 2 , April 2021 , pp. 383 - 398
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

The work of the first author was supported in part by Shanghai Sailing Program (17YF1413800) and the National Natural Science Foundation of China (11701373). The work of the second author was supported in part by the Natural Science Foundation of Hunan Province (2017JJ3085 and 2019JJ50172).

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