Exposed faces and duality for symmetric and unitarily invariant norms
In this series of three papers the author investigates the faces of the unit ball of a symmetric gauge function on \(\mathbb{R}^ n\) and the unit ball of a unitarily invariant norm on the \(n \times m\) matrices. (A symmetric gauge function is also known as a norm invariant under signed permutations of the coordinates.) To study faces of the unit ball of a symmetric gauge function it is essential to study standard faces, i.e. those faces whose relative interior meets the standard cone \(D_ +\) of \(\mathbb{R}^ n\), where \(D_ +\) is the cone consisting of the nonnegative vectors with a decreasing set of coordinates. The trace of a face is the intersection of the face and the standard cone \(D_ +\). The first paper (pp. 349-395; see Zbl 0808.15013) describes the set of symmetries of a face of a symmetric gauge function in terms of its barycenter. Then it characterizes completely standard faces and the subsets of the unit ball that are traces of faces of the unit ball of a symmetric gauge function. In the second paper (pp. 451-493; see Zbl 0808.15014) the author investigates the faces of the unit ball of a unitarily invariant norm \(\psi\) on the \(n \times m\) matrices. There is a closely related symmetric gauge function \(g\) on \(\mathbb{R}^ n\) associated with \(\psi\), namely the one that satisfies \(\psi (A) = g(s(A))\) where \(s(A)\) is the vector of singular values in decreasing order. This paper gives a complete characterization of the faces for the unit ball for \(\psi\) in terms of the faces of the unit ball for \(g\). The face generated by a single matrix is characterized. The third paper (pp. 429-450) investigates the relation between the exposed faces of the unit ball for a unitarily invariant norm \(\psi\) on the \(n \times m\) matrices and those for the corresponding dual norm. The subdifferential of \(\psi\) is completely determined by the subdifferential of \(g\).
scientific article; zbMATH DE number 553971
scientific article