Doctor of Philosophy
Faculty of Informatics
Bray, Brian Michael, Invariant measures on multifids of convex polytopes, Doctor of Philosophy thesis, Faculty of Informatics, University of Wollongong, 2009. http://ro.uow.edu.au/theses/3403
Some convex polytopes are studied from the viewpoint of Geometric Probabilities. The probabilities calculated in this thesis all involve -randomness. This type of randomness is the only form based on invariant measures. Under this type of randomness, if a convex body, & is contained in another convex body ., the probability of hitting &, given that it hits the parent body does not depend on the orientation or position of &. This property reduces the complexity of the mathematics. In two dimensions this invariant measure is length, leading to calculations involving the perimeters of various embedded polygons. In three dimensions the invariant measure is the integral of mean curvature M, so the calculations involve only lengths and dihedral angles. Here, the polyhedra encountered are of such variety that new nomenclature was required. Initially, the cicatrices and convex hulls were used to calculate various probabilities. As the work evolved, the roles were reversed; the cicatrices and hulls became more interesting than the probabilities. Eventually the valuations in three-dimensions were related back to valuations of embedded polytopes, so that measures also usurped probabilities. Key words: -randomness, Integrals of Mean Curvature, Fistula, Convex Hull, Cicatrix, Multifid, Fracture Function, Gleichzeitig Function, Pyramoid, Fastigium.