Radii minimal projections of polytopes and constrained optimization of symmetric polynomials

René Brandenberg 1  and Thorsten Theobald 2
  • 1 Zentrum Mathematik, Technische Universität Mu"nchen, Boltzmannstr. 3, D-85747 Garching bei München.
  • 2 Institut für Mathematik, MA 6-2, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin.

Abstract

We provide a characterization of the radii minimal projections of polytopes onto j-dimensional subspaces in Euclidean space . Applied to simplices this characterization allows to reduce the computation of an outer radius to a computation in the circumscribing case or to the computation of an outer radius of a lower-dimensional simplex. In the second part of the paper, we use this characterization to determine the sequence of outer (n – 1)-radii of regular simplices (which are the radii of smallest enclosing cylinders). This settles a question which arose from an error in a paper by Weißbach (1983). In the proof, we first reduce the problem to a constrained optimization problem of symmetric polynomials and then to an optimization problem in a fixed number of variables with additional integer constraints.

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Advances in Geometry is a mathematical journal which publishes original research articles of excellent quality in the area of geometry. Geometry is a field of long-standing tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity, and geometric ideas and the geometric language permeate all of mathematics.

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