We know in ROCKAFELLAR's convex analysis chap 10 that the collection of convex functions on compact set is sequentially compact. I wonder if it is still true for the collection of 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 $$\mathcal{G}=\{ h:\Delta^n\to\mathbb{R} \mid h(\pi)\leq h(\pi') \text{ if } \pi \preceq \pi'\}.$$

Is $\mathcal{G}$ sequentially compact?

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?