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Tight frames and related geometric problems

Part of: Basic linear algebra General convexity Manifolds

Published online by Cambridge University Press:  18 December 2020

Grigory Ivanov
Grigory Ivanov*
Affiliation:
Institute of Science and Technology Austria (IST Austria), Am Campus 1, Klosterneuburg3400, Austria Laboratory of Combinatorial and Geometrical Structures, Moscow Institute of Physics and Technology, Moscow141701, Russia
*
e-mail: [email protected]
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Abstract

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.

Keywords

Tight frameGrassmannianzonotopevolume

MSC classification

Primary: 15A45: Miscellaneous inequalities involving matrices 52A38: Length, area, volume 49Q20: Variational problems in a geometric measure-theoretic setting 52A40: Inequalities and extremum problems
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 942 - 963
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

The author was supported by the Swiss National Science Foundation grant 200021_179133. The author acknowledges the financial support from the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant no. 075-15-2019-1926.

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