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?



Is this function well studied?
Yes, this function is well-studied. It can be maximized by semidefinite programming (and is in fact one of the standard examples of a non-trivial function that can be maximized by SDP).


The paper "[M. Huhtanen and O. Seiskari. Computational geometry of positive definiteness, Linear Algebra and its Applications, 437:1562-1578, 2012](http://math.aalto.fi/~mhuhtane/posdefcomp_rev.pdf)" is devoted entirely to maximizing this function, and based on your previous posts about the joint numerical range, I expect that this is what you want to do?


