# Multiplicity of Minimum Eigenvalue of a Convex Combination of Hermitian matrices?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Consider the problem \begin{align} \lambda^{\star}=\max_{}&\lambda_{min}\left(\sum_{i=1}^{L}r_iA_i\right) \\ &r_i\geq 0,~\sum_{i=1}^{L}r_i=1 \end{align} $\lambda_{min}(.)$ denotes the minimum eigenvalue of the argument matrix. I am familiar that this is a convex problem and am familiar with the tools to solve it. I am interested in studying the multiplicity of this minimum eigenvalue. Any pointers to literature related to such problems?

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