v2006.03.09 - Convex Optimization

41Figure 7: These bees construct a honeycomb by solving a convex optimizationproblem. (4.4.2.2.3) The most dense packing of identical spheres about acentral sphere in two dimensions is 6. Packed sphere centers describe aregular lattice.The EDM is studied in chapter 4, Euclidean distance matrix, itsproperties and relationship to both positive semidefinite and Gram matrices.We relate the EDM to the four classical properties of the Euclidean metric;thereby, observing existence of an infinity of properties of the Euclideanmetric beyond the triangle inequality. We proceed by deriving the fifthEuclidean metric property and then explain why furthering this endeavoris inefficient because the ensuing criteria (while describing polyhedra inangle or area, volume, content, and so on ad infinitum) grow linearly incomplexity and number with problem size. Some geometrical problemssolvable via EDMs, EDM problems posed as convex optimization, andmethods of solution are presented. Methods of reconstruction are discussedand applied to the map of the United States; e.g., Figure 6. We also generatea distorted but recognizable isotonic map of the USA using only comparativedistance information (only ordinal distance data).We offer a new proof of the Schoenberg characterization of Euclideandistance matrices in chapter 4;D ∈ EDM N⇔{−VTN DV N ≽ 0D ∈ S N h(479)Our proof relies on fundamental geometry; assuming, any EDM mustcorrespond to a list of points contained in some polyhedron (possibly atits vertices) and vice versa. It is known, but not obvious, this Schoenbergcriterion implies nonnegativity of the EDM entries; proved here.