Given a collection of $n$ matrices $A_i$, we could ask for the $B$ such that:

$$\textrm{Minimize: }\quad \textrm{Tr}[B]$$ $$\textrm{Such that: }\forall_i\, B \succeq A_i$$

Here $\succeq$ is in the sense of positive semidefinite cones, that is, $A \succeq B$ iff $A - B$ is positive semidefinite.

I'm looking for a nice description of $B$ in terms of the $A_i$. When $n=1$, it's simply that $B = A_1$. When $n=2$, the solution can be written in terms of eigendecomposition of $A_1 - A_2$:

$$A_2 + \sum_j \max(\lambda_j,0) v_j v_j^T$$

So that it effectively only sums over the positive eigenvalues. Unfortunately, I couldn't find any similar description for $n > 2$.