756 research outputs found
Radii minimal projections of polytopes and constrained optimization of symmetric polynomials
We provide a characterization of the radii minimal projections of polytopes
onto -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 -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.Comment: Minor revisions. To appear in Advances in Geometr
Sharpening Geometric Inequalities using Computable Symmetry Measures
Many classical geometric inequalities on functionals of convex bodies depend
on the dimension of the ambient space. We show that this dimension dependence
may often be replaced (totally or partially) by different symmetry measures of
the convex body. Since these coefficients are bounded by the dimension but
possibly smaller, our inequalities sharpen the original ones. Since they can
often be computed efficiently, the improved bounds may also be used to obtain
better bounds in approximation algorithms.Comment: This is a preprint. The proper publication in final form is available
at journals.cambridge.org, DOI 10.1112/S002557931400029
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