What is the convex hull of the rank-$k$ PSD matrix? What is the convex hull (convex envelope) of the following non-convex set?$$\left\{X \succeq 0 \mid \mbox{rank}(X) = k \right\}$$
If we further require $X=VV^T$, where $V^T V = I$ and $V$ is $n \times k$. Then the convex hull of the above set is
$$\left\{X \mid 0 \preceq X \preceq I, \mbox{Trace}(X) = k \right\}$$
See this. But for the general rank-$k$ PSD matrix, what is the convex hull?
This is an interesting problem, but it is still open.
 A: If one assumes that  the $X$'s are Hermitian then there is a straightforward answer.
Let $E=\{ X | X \ge 0, X^* = X, \operatorname{rk} X = k \} $,
$G=\{ X | X \ge 0, X^* = X, \operatorname{rk} X \ge k \}$.
Then we have
$\operatorname{co} E = G$.
If $X_k \ge 0$ and $\lambda_k >0$, with $\sum_k \lambda_k = 1$, then
$\ker (\sum_k \lambda_k X_k) \subset \cap_k \ker X_k$. It follows that 
$\operatorname{rk}(\sum_k \lambda_k X_k) \ge k$ and so
$G$ is convex (showing $\sum_k \lambda_k X_k \ge 0$ is straightforward), and since we have
$E \subset G$ then $\operatorname{co} E \subset G$.
Now suppose $X \in G$. Let $r=\operatorname{rk} X$. If $r=k$ then $X \in E$, so suppose $r>k$. We can write
$X=U \Lambda U^*$, where $\Lambda = \operatorname{diag} \{ \lambda_1,...,\lambda_r,0,\cdots, 0 \}$ and $\lambda_i >0$.
We can write $\Lambda = {1 \over 2}\operatorname{diag} \{ \lambda_1,...,\lambda_{r-2},0,2\lambda_r,0,\cdots, 0 \} + {1 \over 2}\operatorname{diag} \{ \lambda_1,...,\lambda_{r-2},2\lambda_{r-1},0,0,\cdots, 0 \}$, so, by induction, it 
should be clear that we can write $\Lambda = \sum_p \mu_p \Lambda_p$,
where $\Lambda_p \in E$ and the $\mu_p$ are convex multipliers. Hence $X = \sum_p \mu_p U\Lambda_pU^* \in 
\operatorname{co} E$.
Depending on your definition, $G$ is, in fact, a convex cone. It does not include the 'point' $0$.
