Strict convexity of the joint c-numerical range

Let H=(H1,H2,…,Hm)H=(H1,H2,…,Hm) be an m-tuple of n × n   hermitian matrices, and c=(c1,c2,…,cn)∈Rnc=(c1,c2,…,cn)∈Rn. The joint c  -numerical range Wc(H)Wc(H) of H   is defined. For x=(x1,x2,…,xm)∈Rmx=(x1,x2,…,xm)∈Rm, the form associated with H is defined asFH(t,x)=det(tIn+x1H1+x2H2+⋯+xmHm).FH(t,x)=det(tIn+x1H1+x2H2+⋯+xmHm).We prove that the convex hull of Wc(H)Wc(H) is strictly convex if the roots of FH(t,x)=0FH(t,x)=0 have constant multiplicities for all x   on the unit sphere in RmRm. The result is illustrated with several examples.