Une fonction sur qui est -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 , à la notion de valeur singulière signée.
A function on which is -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 , to the notion of signed singular value.
@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}, volume = {Ser. 6, 16}, number = {1}, year = {2007}, doi = {10.5802/afst.1139}, mrnumber = {2325592}, zbl = {pre05247239}, language = {en}, url = {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/
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