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Radii minimal projections of polytopes and constrained optimization of symmetric polynomials

Rene Brandenberg·Thorsten Theobald
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Abstract

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 Weißbach (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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Rene Brandenberg, Thorsten Theobald. 2005-10-11. Radii minimal projections of polytopes and constrained optimization of symmetric polynomials. https://arxiv.org/abs/math/0311017

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