Distance to the convex hull of the unitary orbit with respect to unitary similarity invariant norms

Let A B be Hermitian matrices. Denote by co the convex hull of the unitary orbit of B. We determine the matrices BM and Bm in co such that for all X∊ co and all unitary similarity invariant norms ∣∣sdot;∣∣, ie., those norms ∣∣sdot;∣∣ satisfying ∣∣UXU ∗∣∣=∣∣X∣∣ for any Hermitian X and unitary U. As a result, we can easily compute the quantities and A particular case of these results answers a question raised by Dr. Y. Nakamura.