Convex <span class="mathjax-formula">$\operatorname{SO}(N)\times \operatorname{SO}(n)$</span>-invariant functions and refinements of von Neumann’s inequality

Dacorogna, Bernard; Maréchal, Pierre

doi:10.5802/afst.1139

Convex

SO (N) \times SO (n)

-invariant functions and refinements of von Neumann’s inequality

Dacorogna, Bernard ; Maréchal, Pierre

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 16 (2007) no. 1, pp. 71-89.

Résumé
Abstract

Une fonction $f$ sur $M_{N \times n} (ℝ)$ qui est $SO (N) \times SO (n)$ -invariante est convexe si et seulement si sa restriction au sous-espace des matrices diagonales est convexe. Ceci résulte de variantes de l’inégalité de Von Neumann et fait appel, dans le cas où $N = n$ , à la notion de valeur singulière signée.

A function $f$ on $M_{N \times n} (ℝ)$ which is $SO (N) \times SO (n)$ -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.

MR 2325592

DOI : https://doi.org/10.5802/afst.1139

@article{AFST_2007_6_16_1_71_0,
     author = {Dacorogna, Bernard and Mar\'echal, Pierre},
     title = {Convex $\operatorname{SO}(N)\times \operatorname{SO}(n)$-invariant functions and refinements of von Neumann's inequality},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {71--89},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 16},
     number = {1},
     year = {2007},
     doi = {10.5802/afst.1139},
     mrnumber = {2325592},
     language = {en},
     url = {http://archive.numdam.org/item/AFST_2007_6_16_1_71_0/}
}

Dacorogna, Bernard; 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, Série 6, Tome 16 (2007) no. 1, pp. 71-89. doi : 10.5802/afst.1139. http://archive.numdam.org/item/AFST_2007_6_16_1_71_0/

Bibliographie

[1] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Archives For Rational Mechanics and Analysis, 63, p. 337-403 (1977). | MR 475169 | Zbl 0368.73040

[2] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, 1989. | MR 990890 | Zbl 0703.49001

[3] B. Dacorogna, P. Marcellini, Implicit Partial Differential Equations, Birkhäuser, 1999. | MR 1702252 | Zbl 0938.35002

[4] B. Dacorogna, H. Koshigoe, On the different notions of convexity for rotationally invariant functions, Annales de la Faculté des Sciences de Toulouse, II(2), p. 163-184 (1993). | Numdam | MR 1253387 | Zbl 0828.49016

[5] J.-B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Minimization Algorithms, I and II, Springer-Verlag, 1993. | MR 1261420

[6] R. A. Horn, C. A. Johnson, Matrix Analysis, Cambridge University Press, 1985. | MR 832183 | Zbl 0576.15001

[7] B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Annales Scientifiques de l’Ecole Normale Supérieure, 6, p. 413-455 (1973). | Numdam | MR 364552 | Zbl 0293.22019

[8] P. J. Laurent, Approximation et Optimisation, Hermann, 1972. | MR 467080 | Zbl 0238.90058

[9] H. Le Dret, Sur les fonctions de matrices convexes et isotropes, Comptes Rendus de l’Académie des Sciences, Paris, Série 1, Mathématiques, 310, p. 617-620 (1990). | MR 1050144 | Zbl 0693.15013

[10] A. Lewis, Group invariance and convex matrix analysis, SIAM Journal of Matrix Analysis and Applications, 17, p. 927-949 (1996). | MR 1410709 | Zbl 0876.15016

[11] A. Lewis, Convex analysis on Cartan subspaces, Nonlinear Analysis, 42, p. 813-820 (2000). | MR 1776925 | Zbl 01513297

[12] A. Lewis, The mathematics of eigenvalue optimization, Mathematical Programming, Series B 97, p. 155-176 (2003). | MR 2004395 | Zbl 1035.90085

[13] P. Rosakis, Characterization of convex isotropic functions, Journal of Elasticity, 49, p. 257-267 (1997). | MR 1633494 | Zbl 0906.73018

[14] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970. | MR 274683 | Zbl 0193.18401

[15] A. Seeger, Convex analysis of spectrally defined matrix functions, SIAM Journal on Optimization, 7(3), p. 679-696 (1997). | MR 1462061 | Zbl 0890.15018

[16] D. Serre, Matrices: Theory and Applications, Grad. Text in Math. 216, Springer-Verlag, 2002. See also http://www.umpa.ens-lyon.fr/ $\sim$ serre/publi.html. | MR 1923507 | Zbl 1011.15001

[17] F. Vincent, Une note sur les fonctions convexes invariantes, Annales de la Faculté des Sciences de Toulouse, p. 357-363 (1997). | Numdam | MR 1611773 | Zbl 0915.17007