# Is this function well studied?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Define the simplex \begin{align} \mathcal{S}=\left\{[x_1,\dots,x_L]\mid x_i\geq 0,~\sum_{i=1}^{L}x_i=1 \right\} \end{align} and consider the function over the simplex \begin{align} f(x_1,\dots,x_L)=\lambda_{min}\left(\sum_{i=1}^{L}x_iA_i\right) \end{align} where $\lambda_{min}(.)$ denotes the minimum eigenvalue. What are the properties of this function?

This is a concave function (the infimum of the affine functions $\sum_{i} x_i \langle v, A_i v \rangle$ for $v$ in the unit sphere).