Semiconvexity of invariant functions of rectangular matrices

M. Silhavý
May 2003
Calculus of Variations & Partial Differential Equations;May2003, Vol. 17 Issue 1, p75
Academic Journal
Abstract. The paper deals with the semiconvexity properties (i.e., the rank 1 convexity, quasiconvexity, polyconvexity, and convexity) of rotationally invariant functions f of $m/FORMULA> matrices. For $m\neq n$ the invariance with respect to the proper orthogonal group and the invariance with respect to the full orthogonal group coincide. With each invariant f one can associate a fully invariant function $f_\flat$ of a square matrix of type $p\times p$ where $p = \min \{m,n\}.$ It is shown that f has the semi convexity of a given type if and only if $f_\flat$ has the semiconvexity of that type. Consequently the semiconvex hulls of f can be determined by evaluating the corresponding hulls of $f_\flat$ and then extending them to $m/FORMULA> matrices by rotational invariance.


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