Local approaches to the isoperimetric problem for octahedra

Abstract

In this thesis, we study the isoperimetric problem for octahedra. By Lindelof-Minkowski's theorem, we can reduce our problem to the study of circumscribing octahedra of a sphere. Using a volume formula by Hsiang for tangent polyhedron, we obtain a gradient formula and a formula of quadratic form for the volume function of octahedra. By applying these two formulae in mathematica, we obtain a local minimum of the volume function with precision of 10[supersprite -8]. In Chapter 1, we introduce the isoperimetric problem, including a short history, the solved and unsolved problems. Then we discuss how to reduce our problem to the cases of circumscribing polyhedra. Finally we introduce the concept of spherical configuration. In Chapter 2, we introduce a clean-cut volume formula for tangent polyhedra, and some of its applications. In Chapter 3, we derive a gradient formula and a formula for quadratic form of the volume function. Then we describe a concise method for determinating a local minimum of the volume function. In Chapter 4, we give discussions and future works.

Date of Award	2001
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology