Hello, considering that for real numbers, the intersection of intervals defined by simple inequalities has a quite simple form as $$ \bigcap_i\{x|x\leq a_i\}=\{x|x\leq\min_i\{a_i\}\} $$

However, what is the case if the variables are chosen as Hermitian matrices, and the intersection defined by inequality is replaced with the convex cone defined by the generalized inequality?

All variables in following are assumed to be Hermitian matrices.

To be specific, define the generalize inequality $X\preceq A_i$ to denote that $X-A_i$ is negative semi-definite, then $\{X|X\preceq A_i\}$ defines a convex cone in the Hermitian matrix space.

Is there any result about the intersection of these cones? To say, can the following set be simplified? $$ \bigcap_i\{X|X\preceq A_i\} $$

When does there exist such an $A$ to satisfy $\{X|X\preceq A\}=\bigcap_i\{X|X\preceq A_i\}$?

Any suggestion or comment on this question will be appreciated and thanks very much for your help!