On the convex hull of the multiform numerical range

Abstract

Let A 1,…,Am be nxn hermitian matrices. Definine W(A 1,…,Am )={(xA1x ∗,…xAmx ∗):x∊C n ,xx ∗=1}. We will show that every point in the convex hull of W(A 1,…,Am ) can be represented as a convex combination of not more than k(m,n) points in W(A 1,…,Am ) where k(m,n)=min{n,[√m]+δ n 2 m+1}.