Suppose we are given $P_1,\cdots,P_n \in \mathbb{C}^{m\times m}$ as semi-definite positive semidefinite matrices.
How to characterize the set $S$ of their common lower bounds, $$S=\{Q|0\leq Q\leq P_i, for ~all~ i\}，$$ where $A\leq B$ means $B-A$ is semi-definite positive$P_1, P_2, \dots, P_n \in \mathbb{C}^{m \times m}$.
The set is a convex set, how to describe all the extreme points.
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?
