Radii minimal projections of polytopes and constrained optimization of symmetric polynomials
Radii minimal projections of polytopes and constrained optimization of symmetric polynomials
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多面体的半径最小投影和对称多项式的约束优化
DOI：
10.1515/advgeom.2006.005
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
T. Theobald
中科院分区：
文献类型：
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作者：
René Brandenberg;T. Theobald
我们提供了多面体在欧几里德空间 $\E^n$ 中 $j$ 维子空间上的半径最小投影的表征。应用于单纯形时，该特征允许将外半径的计算减少为外接情况下的计算或降低为低维单纯形的外半径的计算。在本文的第二部分中，我们使用此特征来确定正则单纯形的外部 $(n-1)$ 半径序列（它们是最小包围圆柱体的半径）。这解决了由 Wei{\ss}bach (1983) 的一篇关于这一决定的论文是错误的事件引起的问题。在证明中，我们首先将问题简化为对称多项式的约束优化问题，然后简化为具有附加整数约束的固定数量变量的优化问题。
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.