Radii minimal projections of polytopes and constrained optimization of symmetric polynomials

Details Radii minimal projections of polytopes and constrained optimization of symmetric polynomials Radii minimal projections of polytopes and constrained optimization of symmetric polynomials

2003-11-03

arxiv.org/abs/math/0311017v3

Mathematics

Metric Geometry

Minor revisions. To appear in Advances in Geometry

Scientific paper

We provide a characterization of the radii minimal projections of polytopes onto $j$-dimensional subspaces in Euclidean space $\E^n$. Applied on 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 the incidence that a paper by Wei{\ss}bach (1983) on this determination was erroneous. 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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