Suppose we are given $P_1,\cdots,P_n \in \mathbb{C}^{m\times m}$ as semi-definite positive 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.

The set is a convex set, how to describe all the extreme points.