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We know in ROCKAFELLAR's convex analysis chap 10 that the collection of uniformly bounded convex functions on compact set is sequentially compact. I wonder if it is still true for the collection of bounded Schur convex functions. Specifically, let $$\Delta^n=\{\mathbf{\pi} \in \mathbb{R}^n \mid \pi_i\geq0, \sum_{i=1}^n \pi_i=1 \} $$ and $$\mathcal{G}=\{ h:\Delta^n\to\mathbb{R} \mid |h| \leq C, h(\pi)\leq h(\pi') \text{ if } \pi \preceq \pi'\}.$$

Is $\mathcal{G}$ sequentially compact?

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  • $\begingroup$ sry, forgot the uniformly boundedness. editted. $\endgroup$
    – Sung-En Chiu
    Commented Apr 20, 2017 at 2:29

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