Distances from a Hermitian pair to diagonalizable and nondiagonalizable Hermitian pairs

Let W(T) and r(T) denote the numerical range and numerical radius of an n x n complex matrix T. Let H-n(2) denote the space of pairs of n x n Hermitian matrices. Define a norm on H-n(2) by parallel to(X, Y)parallel to = r(X + iY). Take (A, B) is an element of H-n(2). It is shown that if 0 is an element of W(A + iB), then inf{\mu\ : mu is not an element of W(A + iB)} is an upper bound on the distance to the nearest pair that is simultaneously diagonalizable by congruence. If 0 is not an element of W(A + iB), then min{\mu\ : mu is an element of W(A + iB)}, which is the Crawford number of the pair (A, B), is equal to the distance to the nearest pair that is not simultaneously diagonalizable by congruence. The results are similar when the numerical radius is replaced by the spectral norm.