An approximation problem with respect to symmetric gauge functions with application to matrix spaces

Abstract

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 y_M and y_m 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 B_M and B_m in convB/∼, the convex hull of B/∼{XεM:X∼B}, such that

for all XεconvB/∼ and all ∼-invariant norms.

^*The author wishes to express his thanks to Dr. Yik-Hoi Au-Yeung for hi valuable advice and encouragement.

^*The author wishes to express his thanks to Dr. Yik-Hoi Au-Yeung for hi valuable advice and encouragement.

Notes

^*The author wishes to express his thanks to Dr. Yik-Hoi Au-Yeung for hi valuable advice and encouragement.