set of products of rank-k matrices

Question

Derive the convex hull of the set of rank k outer products: $\mathcal{D}=\{XX^T:X\in\Re^{b\times k},\text{rank}(X)=k\}$.

Question:

By definition, $\text{Conv}(\mathcal{D})=\{\sum_{i=1}^n\alpha_iX_iX_i^T:X_i\in\mathcal{D},\forall i=1,\cdots,n;\forall n\in N\}$.

Is there some descriptions or characterization of elements in $\text{Conv}(\mathcal{D})$? I cannot see it.

I can see the element in $\text{Conv}(\mathcal{D})$ should be positive semi-definite. Anything else?

Thank you!

Answer 1

Hint: It suffices to observe the following:

• If $A,B$ are positive semidefinite, then $\operatorname{rank}(A + B) \geq \operatorname{rank}(A)$
• For $r \geq 2$, any rank-$r$ matrix can be written in the form $\frac 12 (A + B)$ for some choice of positive semidefinite matrices $A,B$ of rank $r-1$.