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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| Published in | Canadian mathematical bulletin Vol. 64; no. 4; pp. 942 - 963 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Canada
Canadian Mathematical Society
01.12.2021
Cambridge University Press |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0008-4395 1496-4287 |
| DOI | 10.4153/S000843952000096X |
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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. |
|---|---|
| AbstractList | 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. 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. |
| Author | Ivanov, Grigory |
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| Cites_doi | 10.1112/blms/16.3.278 10.1007/978-1-4613-9425-9 10.1112/S0025579300001418 10.1007/978-3-0348-5858-8_13 10.4310/jdg/1102536713 10.1007/BFb0090058 10.1007/BF02187792 10.1090/S0002-9947-1990-0989573-6 10.1090/surv/223 10.4153/CJM-1974-032-5 10.1016/0022-1236(88)90068-7 10.1007/978-3-0348-8272-9_20 10.1007/BF02787224 10.1007/s00454-001-0066-3 10.1017/CBO9780511543173 10.1007/978-3-0348-0439-4_9 10.2140/pjm.1979.83.543 |
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| References_xml | – volume: 317 start-page: 611 issue: 2 year: 1990 end-page: 629 article-title: The extreme projections of the regular simplex publication-title: Trans. Am. Math. Soc – volume: 26 start-page: 302 issue: 2 year: 1974 end-page: 321 article-title: Combinatorial properties of associated zonotopes publication-title: Canad. J. Math – volume: 83 start-page: 543 issue: 2 year: 1979 end-page: 553 article-title: A geometric inequality with applications to linear forms publication-title: Pac. J. Math – volume: 16 start-page: 278 issue: 3 year: 1984 end-page: 280 article-title: Volumes of projections of unit cubes publication-title: Bull. Lond. Math. Soc – volume: 5 start-page: 93 issue: 2 year: 1958 end-page: 102 article-title: Some extremal problems for convex bodies publication-title: Mathematika – volume: 64 start-page: 207 issue: 2 year: 1988 end-page: 228 article-title: The largest projections of regular polytopes publication-title: Isr. J. Math – volume: 27 start-page: 251 issue: 3 year: 1988 end-page: 262 article-title: Extremum problems for zonotopes publication-title: Geom. Dedi – volume: 63 start-page: 1 issue: 2 year: 2017 end-page: 5 article-title: On the volume of the John–Löwner ellipsoid publication-title: Discrete Comp. Geom – volume: 5 start-page: 305 issue: 3 year: 1990 end-page: 322 article-title: Exterior algebra and projections of polytopes publication-title: Discrete Comput. Geom – volume: 21 start-page: 103 issue: 1–2 year: 1986 end-page: 110 article-title: The measures of the projections of a cube publication-title: Studia Sci. Math. Hungar – volume: 27 start-page: 215 issue: 2 year: 2002 end-page: 226 article-title: Hyperplane projections of the unit ball of ℓp n publication-title: Discrete Comput. Geom – volume: 80 start-page: 109 issue: 1 year: 1988 end-page: 123 article-title: Sections of the unit ball of ℓp n publication-title: J. Funct. Anal – volume: 68 start-page: 159 issue: 1 year: 2004 end-page: 184 article-title: Volume inequalities for subspaces of Lp publication-title: J. Differ. Geom – ident: S000843952000096X_r13 doi: 10.1112/blms/16.3.278 – ident: S000843952000096X_r9 doi: 10.1007/978-1-4613-9425-9 – ident: S000843952000096X_r15 doi: 10.1112/S0025579300001418 – ident: S000843952000096X_r16 doi: 10.1007/978-3-0348-5858-8_13 – ident: S000843952000096X_r12 doi: 10.4310/jdg/1102536713 – volume: 21 start-page: 103 year: 1986 ident: S000843952000096X_r4 article-title: The measures of the projections of a cube publication-title: Studia Sci. Math. Hungar – ident: S000843952000096X_r2 doi: 10.1007/BFb0090058 – ident: S000843952000096X_r7 doi: 10.1007/BF02187792 – ident: S000843952000096X_r8 doi: 10.1090/S0002-9947-1990-0989573-6 – ident: S000843952000096X_r1 doi: 10.1090/surv/223 – ident: S000843952000096X_r17 doi: 10.4153/CJM-1974-032-5 – ident: S000843952000096X_r14 doi: 10.1016/0022-1236(88)90068-7 – ident: S000843952000096X_r19 doi: 10.1007/978-3-0348-8272-9_20 – volume: 27 start-page: 251 year: 1988 ident: S000843952000096X_r5 article-title: Extremum problems for zonotopes publication-title: Geom. Dedi – ident: S000843952000096X_r6 doi: 10.1007/BF02787224 – ident: S000843952000096X_r3 doi: 10.1007/s00454-001-0066-3 – ident: S000843952000096X_r20 doi: 10.1017/CBO9780511543173 – ident: S000843952000096X_r11 doi: 10.1007/978-3-0348-0439-4_9 – ident: S000843952000096X_r18 doi: 10.2140/pjm.1979.83.543 – volume: 63 start-page: 1 year: 2017 ident: S000843952000096X_r10 article-title: On the volume of the John–Löwner ellipsoid publication-title: Discrete Comp. Geom |
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| Snippet | 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... 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... |
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| Title | Tight frames and related geometric problems |
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