An approximation problem with respect to symmetric gauge functions with application to matrix spaces
Che‐Man Cheng · Linear and Multilinear Algebra · 1991
Let x,yε R n . Denote by convy/R the convex hull of the set y/R={z:z=yDP for some diagonal orthogonal matrix D and some permutation matrix P}. We determine yM and ym in convy/R, such that for all zεconvy/R and all symmetric gauge function Φ on R n . The result is then applied to some approximation problems in various matrix spaces M with respect to norms ∣⋅∣ that are invariant under certain equivalence relation ∼, i.e., those norms ∣⋅∣ on M satisfying In particular, for any A,BεM, we determine the matrices BM and Bm in convB/∼, the convex hull of B/∼{XεM:X∼B}, such that for all XεconvB/∼ and all ∼-invariant norms.