Geodesic
On semiconvexity properties of rotationally invariant functions in two dimensions
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 559-571

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Let $f$ be a function defined on the set ${\mathbf M}^{2\times 2}$ of all $2$ by $2$ matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function $f$ can be represented as a function $\tilde{f}$ of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of $f$ in terms of its representation $\tilde{f}.$
Classification : 26B25, 49J10, 49J45, 74B20, 74G65
Keywords: semiconvexity; rank 1 convexity; polyconvexity; convexity; rotational invariance 
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     author = {\v{S}ilhav\'y, M.},
     title = {On semiconvexity properties of rotationally invariant functions in two dimensions},
     journal = {Czechoslovak Mathematical Journal},
     pages = {559--571},
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     volume = {54},
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     zbl = {1080.49013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2004__54_3_a1/}
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Šilhavý, M. On semiconvexity properties of rotationally invariant functions in two dimensions. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 559-571. http://geodesic.mathdoc.fr/item/CMJ_2004__54_3_a1/