A Spectral Generalization of Von Neumann Minimax Theorem

Abstract

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.