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Permanent link (DOI): https://doi.org/10.7939/R38G8FX53

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Topics in Convex Geometric Analysis and Discrete Tomography Open Access

Descriptions

Other title
Subject/Keyword
Projection
Congruent
Grünbaum’s Inequality
Lattice Set
Section
Convex Body
Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Zhang, Ning
Supervisor and department
Vladyslav Yaskin
Examining committee member and department
Yaskin, Vladyslav (Mathematical and Statistical Sciences)
Troitsky, Vladimir (Mathematical and Statistical Sciences)
Dai, Feng (Mathematical and Statistical Sciences)
Stancu, Alina (Mathematics and Statistics, Concordia University )
Litvak, Alexander (Mathematical and Statistical Sciences)
Yu, Xinwei (Mathematical and Statistical Sciences)
Department
Department of Mathematical and Statistical Sciences
Specialization
Mathematics
Date accepted
2017-06-27T14:25:36Z
Graduation date
2017-11:Fall 2017
Degree
Doctor of Philosophy
Degree level
Doctoral
Abstract
In this thesis, some topics in convex geometric analysis and discrete tomography are studied. Firstly, let K be a convex body in the n-dimensional Euclidean space. Is K uniquely determined by its sections? There are classical results that explain what happens in the case of sections passing through the origin. However, much less is known about sections that do not contain the origin. Here, several problems of this type and the corresponding uniqueness results are established. We also establish a discrete analogue of the Aleksandrov theorem for the areas and the surface areas of projections. Finally, we find the best constant for the Grünbaum’s inequality for projections, which generalizes both Grünbaum’s inequality, and an old inequality of Minkowski and Radon.
Language
English
DOI
doi:10.7939/R38G8FX53
Rights
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
Citation for previous publication
D. Ryabogin, V. Yaskin, and N. Zhang, Unique determination of convex lattice sets, Discrete Comput. Geom. (3) 57 (2017), 582-589.M. Stephen and N. Zhang, Grünbaum’s inequality for projections, J. Funct. Anal. (6) 272 (2017), 2628–2640.V. Yaskin and N. Zhang, Non-central sections of convex bodies, Israel J. Math.(2007), DOI:10.1007/s11856-017-1532-9.N. Zhang, An analogue of the Aleksandrov projection theorem for convex lattice polygons, Proc. Amer. Math. Soc., 145 (2017), 2305–2310.N. Zhang, On bodies with congruent sections by cones or non-central planes, preprint.

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