A tight frame is the orthogonal projection of some orthonormal basis of $\mathbb {R}^n$ onto $\mathbb {R}^k.$ We show that a set of vectors is a tight frame if and only if the set of all cross products of these vectors is a tight frame. We reformulate a range of problems on the volume of projections (or sections) of regular polytopes in terms of tight frames and write a first-order necessary condition for local extrema of these problems. As applications, we prove new results for the problem of maximization of the volume of zonotopes.

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紧框架和相关几何问题

紧框架是 $\mathbb {R}^n$ 到 $\mathbb {R}^k$的一些正交基的正交投影。 我们证明一组向量是紧框架当且仅当这些向量的所有叉积都是一个紧框架。我们根据紧框架重新表述规则多胞体的投影（或截面）体积的一系列问题，并为这些问题的局部极值写出一阶必要条件。作为应用，我们证明了带位体体积最大化问题的新结果。