# Convex $SO\left(N\right)×SO\left(n\right)$-invariant functions and refinements of von Neumann’s inequality

• [1] EPFL, CH-1015 Lausanne, Switzerland
• [2] Université Paul Sabatier, Institut de mathématiques, F-31062 Toulouse cedex 9, France
• Volume: 16, Issue: 1, page 71-89
• ISSN: 0240-2963

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## Abstract

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A function $f$ on ${M}_{N×n}\left(ℝ\right)$ which is $\mathrm{SO}\left(N\right)×\mathrm{SO}\left(n\right)$-invariant is convex if and only if its restriction to the subspace of diagonal matrices is convex. This results from Von Neumann type inequalities and appeals, in the case where $N=n$, to the notion of signed singular value.

## How to cite

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Dacorogna, Bernard, and Maréchal, Pierre. "Convex $\operatorname{SO}(N)\times \operatorname{SO}(n)$-invariant functions and refinements of von Neumann’s inequality." Annales de la faculté des sciences de Toulouse Mathématiques 16.1 (2007): 71-89. <http://eudml.org/doc/10038>.

@article{Dacorogna2007,
abstract = {A function $f$ on $M_\{N\times n\}(\{\mathbb\{R\}\})$ which is $\mathop \{\mathrm\{SO\}(N)\}\times \mathop \{\mathrm\{SO\}(n)\}$-invariant is convex if and only if its restriction to the subspace of diagonal matrices is convex. This results from Von Neumann type inequalities and appeals, in the case where $N=n$, to the notion of signed singular value.},
affiliation = {EPFL, CH-1015 Lausanne, Switzerland; Université Paul Sabatier, Institut de mathématiques, F-31062 Toulouse cedex 9, France},
author = {Dacorogna, Bernard, Maréchal, Pierre},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {singular values; von Neumann's inequality; convex -invariant functions},
language = {eng},
number = {1},
pages = {71-89},
publisher = {Université Paul Sabatier, Toulouse},
title = {Convex $\operatorname\{SO\}(N)\times \operatorname\{SO\}(n)$-invariant functions and refinements of von Neumann’s inequality},
url = {http://eudml.org/doc/10038},
volume = {16},
year = {2007},
}

TY - JOUR
AU - Dacorogna, Bernard
AU - Maréchal, Pierre
TI - Convex $\operatorname{SO}(N)\times \operatorname{SO}(n)$-invariant functions and refinements of von Neumann’s inequality
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2007
PB - Université Paul Sabatier, Toulouse
VL - 16
IS - 1
SP - 71
EP - 89
AB - A function $f$ on $M_{N\times n}({\mathbb{R}})$ which is $\mathop {\mathrm{SO}(N)}\times \mathop {\mathrm{SO}(n)}$-invariant is convex if and only if its restriction to the subspace of diagonal matrices is convex. This results from Von Neumann type inequalities and appeals, in the case where $N=n$, to the notion of signed singular value.
LA - eng
KW - singular values; von Neumann's inequality; convex -invariant functions
UR - http://eudml.org/doc/10038
ER -

## References

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17. F. Vincent, Une note sur les fonctions convexes invariantes, Annales de la Faculté des Sciences de Toulouse, p. 357-363 (1997). Zbl0915.17007MR1611773

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