Suppose we are given positive semidefinite matrices $P_1, P_2, \dots, P_n \in \mathbb{C}^{m \times m}$. 

- How to characterize the set $S$ of their common lower bounds $$S = \{Q \mid 0 \leq Q\leq P_i, \forall i\}$$ where $A \leq B$ means $B-A$ is positive semidefinite? 

- The set is convex. How to describe all the extreme points?