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Six mathematical gems from the history of Distance Geometry

Leo Liberti(leoliberti***at***gmail.com)
Carlile Lavor(clavor***at***ime.unicamp.br)

Abstract: This is a partial account of the fascinating history of Distance Geometry. We make no claim to completeness, but we do promise a dazzling display of beautiful, elementary mathematics. We prove Heron's formula, Cauchy's theorem on the rigidity of polyhedra, Cayley's generalization of Heron's formula to higher dimensions, Menger's characterization of abstract semi-metric spaces, a result of Goedel on metric spaces on the sphere, and Schoenberg's equivalence of distance and positive semidefinite matrices, which is at the basis of Multidimensional Scaling.

Keywords: Euler's conjecture, Cayley-Menger determinants, Multidimensional scaling, Euclidean Distance Matrix

Category 1: Global Optimization (Other )

Category 2: Combinatorial Optimization (Other )

Category 3: Linear, Cone and Semidefinite Programming (Other )

Citation:

Download: [PDF]

Entry Submitted: 02/10/2015
Entry Accepted: 02/15/2015
Entry Last Modified: 02/10/2015

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