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Let $n\geq2$. Is it true that the minimax problem: $$ \min_{p\in\mathcal{P}}\max_{H\in\mathcal H}p^tH\bar{p}, $$ where $\mathcal H\subset\mathcal{M}(n)$ is a strictly convex bounded subset of hermitian positive semi-definite matrices, $\mathcal{P}$ is a hyperplane of $\mathbb{C}^n$ not containing 0, has a unique solution ?

Is there a standard reference for general results about the uniqueness of a solution of minimax problems ?
Thank you.

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  • $\begingroup$ maybe you want to normalize it, or to bound p to the unit sphere? Otherwise the minimum should be 0. (And maybe $\mathcal H$ should be bounded). $\endgroup$ Commented Feb 28, 2022 at 12:05
  • $\begingroup$ Indeed, $\mathcal{H}$ is bounded, and the hyperplane $\mathcal{P}$ does not contain the origin. $\endgroup$ Commented Feb 28, 2022 at 12:14

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