euclidean distance matrix

Six mathematical gems from the history of Distance Geometry

  • Carlile Lavor
  • Leo Liberti
    • Categories Combinatorial Optimization, Global Optimization, Linear, Cone and Semidefinite Programming Tags , , ,

    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 … Read more

    The Gram dimension of a graph

  • Monique Laurent
  • Antonios Varvitsiotis
    • Categories Graphs and Matroids, Linear, Cone and Semidefinite Programming Tags , , , ,

    The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner products on the edges of the graph. The class of graphs satisfying $\gd(G) … Read more