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Permanent link (DOI): https://doi.org/10.7939/R3B09C

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ON SECTIONS OF CONVEX BODIES IN HYPERBOLIC SPACE Open Access

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Other title
Subject/Keyword
Fourier Analysis
Busemann-Petty problem
Geometric Tomography
Convex Geometry
Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Hiripitiyage, Kasun L.H.
Supervisor and department
Yaskin, Vladyslav (Department of Mathematical and Statistical Sciences)
Examining committee member and department
Bouchard, Vincent (Department of Mathematical and Statistical Sciences)
Guay, Nicolas (Department of Mathematical and Statistical Sciences)
Han, Bin (Department of Mathematical and Statistical Sciences)
Department
Department of Mathematical and Statistical Sciences
Specialization
Mathematics
Date accepted
2015-08-28T13:27:49Z
Graduation date
2015-11
Degree
Master of Science
Degree level
Master's
Abstract
The Busemann-Petty problem asks the following: if 𝐾,𝐿 ⊂ ℝⁿ are origin-symmetric convex bodies such that volₙ₋₁(𝐾 ∩ ξ^⊥)) ≤ volₙ₋₁(𝐿 ∩ ξ^⊥) ∀ ξ ∈ Sⁿ⁻¹, is it necessary that volₙ(𝐾) ≤ volₙ(𝐿)? This problem received a lot of attention, and many analogues have been considered. For origin-symmetric convex bodies 𝐾 and 𝐿 in hyperbolic space ℍⁿ, we find a suitable condition which guarantees volₙ(𝐾) ≤ volₙ(𝐿). Origin-symmetry is important in many problems in convex geometry. By Brunn's Theorem, each central hyperplane section of an origin-symmetric convex body 𝐾 ⊂ ℝⁿ has maximal volume amongst all parallel sections of 𝐾. Makai, Martini and Ódor proved the converse of this statement for star bodies. Again working in ℍⁿ, we prove an analogue of this result.
Language
English
DOI
doi:10.7939/R3B09C
Rights
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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