Let $A$ and $B$ be two matrices of eigenvalues $\lambda_i$ and $\mu_i$, respectively.

The *spectral variation* of $B$ w.r.t. $A$ and the *eigenvalue variation* of $B$ and $A$ are, respectively,
\begin{align} s_B(A)&=\max_i\min_j\vert\lambda_i-\mu_j\vert, \\
v(A,B)&=\min_{\pi}\max_i\vert\lambda_i-\mu_{\pi(i)}\vert;\end{align}
where in the latter the minimum is to be taken over all permutations $\pi$ of the indices.

Question 1.If $A$ and $B$ are Hermitian matrices, then for which norms is this true? $$s_B(A)\leq\Vert A-B\Vert.$$

Question 2.If $A$ and $B$ are normal matrices (more generally for fully symmetric operators), then for which norms is this true? $$v(A,B)\leq\Vert A-B\Vert.$$

I would appreciate any reference to the state-of-the-art in this matter.