ON SECTIONS OF CONVEX BODIES IN HYPERBOLIC SPACE Open Access
- Other title
- Type of item
- 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 of Mathematical and Statistical Sciences
- Date accepted
- Graduation date
Master of Science
- Degree level
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.
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