This is related to so-called *hyperbolic polynomials*, studied by L. Gaarding in the fifties. More generally, let $\lambda(\xi)$ be the least eigenvalue of $A(\xi)=\sum_\alpha\xi_\alpha A^\alpha$, where $A^\alpha$ are Hermitian matrices and $\xi$ is a real vector. Then $\lambda$ is a concave function. It is generically strictly concave, except in the radial directions of course because of homogeneity $\lambda(s\xi)=s\lambda(\xi)$ for $s>0$. The strict concavity is related to the lack of commutativity of pairs $(A^\alpha,A^\beta)$. For instance, the Pauli matrices yield $\lambda(\xi)=-|\xi|$, which is clearly strictly concave away from rays ${\mathbb R}^+\xi$. On the opposite, if $[A^\alpha,A^\beta]=0$ for every pair, then $\lambda$ is piecewise linear.

Strict concavity occurs for instance when the least eigenvalue is simple for every $\xi\ne0$, or if it has a constant multiplicity. Then $\lambda$ is analytic away of the origin, with a Hessian matrix of rank $n-1$. It turns out that this property implies a so-called *Strichartz inequality* for the solutions of the system of Partial Differential Equations
$$\frac{\partial u}{\partial t}+\sum_\alpha A^\alpha\frac{\partial u}{\partial x_\alpha}=0.$$
Obviously, the symbol of the system is $\det(\tau I_n+A(\xi))$.Thus the characteristic manifold is related to the eigenvalues of $A(\xi)$, in particular to $\lambda(\xi)$.

The other eigenvalues satisfy more involved inequalities such as Weyl, Lidskii, Ky Fan -type inequalities. For instance, the sum of the $k$ least eigenvalues is concave too. This is a part of Alfred Horn's conjecture, now a theorem thanks to the work of many people, including Knutson & Tao.

The last part of the question, that about Toeplitz-Hausdorff theorem, is unclear. ${\mathbb S}$ is not a singleton, so what means
\begin{align}
\lambda(t)=(1-t)x_1+tx_2, ~~[x_1,x_2]\in \mathbb{S} \qquad ?
\end{align}
The equality certainly holds true for teh particular point $(x_1,x_2)$ obtained by taking $x$ a unit eigenvector associated with $\lambda(t)$, but what else ? To see some deep relations between Toeplitz-Hausdorff and hyperbolic polynomials, have a look to our paper with Th. Gallay, *Numerical measure of a complex matrix*, in
Comm. Pure Appl. Math. **65** (2012), no. 3, 287–336.