Tight frames and related geometric problems

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...

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Bibliographic Details
Published inCanadian mathematical bulletin Vol. 64; no. 4; pp. 942 - 963
Main Author Ivanov, Grigory
Format Journal Article
LanguageEnglish
Published Canada Canadian Mathematical Society 01.12.2021
Cambridge University Press
Subjects
Approximation
Polytopes
volume
52A40
15A45
Tight frame
Grassmannian
52A38
zonotope
49Q20
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ISSN0008-4395
1496-4287
DOI10.4153/S000843952000096X

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Summary: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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ISSN:0008-4395
1496-4287
DOI:10.4153/S000843952000096X