ON SECTIONS OF CONVEX BODIES IN HYPERBOLIC SPACE

• Author / Creator

  Hiripitiyage, Kasun L.H.

• 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.
  

• Subjects / Keywords

  • Busemann-Petty problem
  • Fourier Analysis
  • Convex Geometry
  • Geometric Tomography

• Graduation date

  Fall 2015

• Type of Item

  Thesis

• Degree

  Master of Science

• DOI

  https://doi.org/10.7939/R3B09C

• License

  This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. 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.