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An analogue of the Aleksandrov projection theorem for convex lattice polygons


Author: Ning Zhang
Journal: Proc. Amer. Math. Soc. 145 (2017), 2305-2310
MSC (2010): Primary 05B50, 52C05; Secondary 52B20
DOI: https://doi.org/10.1090/proc/13375
Published electronically: February 15, 2017
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Abstract: Let $ K$ and $ L$ be origin-symmetric convex lattice polytopes in $ \mathbb{R}^n$. We study a discrete analogue of the Aleksandrov projection theorem. If for every $ u\in \mathbb{Z}^n$, the sets $ (K\cap \mathbb{Z}^n)\vert u^\perp $ and $ (L\cap \mathbb{Z}^n)\vert u^\perp $ have the same number of points, is $ K=L$? We give a positive answer to this problem in $ \mathbb{Z}^2$ under the additional hypothesis that $ (2K\cap \mathbb{Z}^2)\vert u^\perp $ and $ (2L\cap \mathbb{Z}^2)\vert u^\perp $ have the same number of points for every $ u\in \mathbb{Z}^n$.


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Additional Information

Ning Zhang
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
Email: nzhang2@ualberta.ca

DOI: https://doi.org/10.1090/proc/13375
Received by editor(s): February 17, 2016
Received by editor(s) in revised form: July 13, 2016
Published electronically: February 15, 2017
Additional Notes: The author was partially supported by a grant from NSERC
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2017 American Mathematical Society