Edge length minimizing polyhedra
Berger, Scott Byron
Hardt, Robert M.
Doctor of Philosophy
This thesis investigates two different edge length minimization problems on convex polyhedra: first, minimizing edge length for fixed volume (Melzak's Problem), and second, minimizing edge length for fixed surface area (Aberth's Problem). Specific examples are given that demonstrate the necessity of restricting the problems to convex polyhedra. The right regular 3-prism of height 1 relative to base edge 1 is shown to minimize Melzak's Problem over several families of polyhedra, including Platonic solids, regular pyramids, and general prisms. For Aberth's Problem, the right regular 3-prism of height 2-13 relative to base edge 1 is shown to minimize over the same families. The minimizer of Aberth's Problem cannot have all equal-area faces; similarly, for Melzak's Problem, the minimizer cannot be an equal-faced polyhedron with 10 or more faces. For the minimizing object in both Aberth's Problem and Melzak's Problem, the area of the kth face must be of the order 1/k 2. The minimizer for Melzak's Problem must exist in a more general class which includes infinite-faced objects. Examples are presented of infinite-faced objects that have finite edge length. Although the minimizer might have an infinite number of faces, the edge skeleton of such a minimizer is proved to consist almost entirely of line segments, which means that the bad points of the edging contribute nothing to the total length of the edge skeleton. Specifically, an edge point is called bad if the local edge set is not a line segment. For a minimizing convex object, the 1-dimensional Hausdorff measure of the set of bad edge points is zero. As a corollary, the edge set of the minimizer for Melzak's Problem does not contain any smooth non-linear arcs.