A Spectral Generalization of Von Neumann Minimax Theorem

Given $n times n$ real symmetric matrices $A_1, dots, A_m$, the following {it spectral minimax} property holds: $$min_{X in mathbf{Delta}_n} max_{y in S_m} sum_{i=1}^m y_iA_i bullet X=max_{y in S_m} min_{X in mathbf{Delta}_n} sum_{i=1}^m y_iA_i bullet X,$$ where $S_m$ is the simplex and $mathbf{Delta}_n$ the spectraplex. For diagonal $A_i$s this reduces to the classic minimax.