Definite triples of Hermitian matrices and matrix polynomials

Let A, B and C be three n×n nonzero Hermitian matrices. The triple (A,B,C) is called definite if the convex hull of the joint numerical range F(A,B,C)={(x∗Ax,x∗Bx,x∗Cx)∈R3: x∈Cn,x∗x=1} does not contain (0,0,0). If the triple (A,B,C) is nondefinite, then the numerical ranges of the matrix polynomials Q(λ)=Aλν3+Bλν2+Cλν1 (ν3>ν2>ν1⩾0) and L(λ)=Aλξ2+(B+i C)λξ1 (ξ2>ξ1⩾0) coincide with the whole complex plane, providing no information. As a consequence, it is of particular interest to characterize a definite triple (A,B,C) and find the distance between (0,0,0) and the boundary of F(A,B,C). The distance between a nondefinite triple (A,B,C) and the “nearest” definite triples with specified properties is also investigated. Moreover, applications of definite triples on matrix polynomials of special interest are presented.