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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 6, Issue 1


Radii minimal projections of polytopes and constrained optimization of symmetric polynomials

René Brandenberg / Thorsten Theobald
Published Online: 2006-05-08 | DOI: https://doi.org/10.1515/ADVGEOM.2006.005


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.

Key Words: Polytope; Projection; outer radius; enclosing cylinder; regular simplex; polynomial optimization; symmetric polynomials

About the article

Received: 2004-04-24

Revised: 2004-10-15

Published Online: 2006-05-08

Published in Print: 2006-01-26

Citation Information: Advances in Geometry, Volume 6, Issue 1, Pages 71–83, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/ADVGEOM.2006.005.

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