Reduction of joint c-numerical ranges

Let (H1,H2,…,Hm)(H1,H2,…,Hm) be an m-tuple of n-by-n   Hermitian matrices and c∈Rnc∈Rn. We investigate the reduction problem of the joint c  -numerical range Wc(H1,H2,…,Hm)Wc(H1,H2,…,Hm). In particular, for m=3m=3 and n⩾3n⩾3, we prove that Wc(H1,H2,H3)Wc(H1,H2,H3) is attainable by the joint numerical range W(K1,K2,K3)W(K1,K2,K3) for some triple Hermitian matrices (K1,K2,K3)(K1,K2,K3).