TITLE: Convex hull of the Stiefel manifold with non-negativity constraints
QUESTION [11 upvotes]: Consider the Stiefel manifold
$$\mathrm{St}(n,k) :=\{X \in \mathbb{R}^{n\times k} : X^TX = I_k\},$$
where $I_k$ is the $k$-dimensional identity matrix. It is well known that 
$$\mathrm{conv} \left( \mathrm{St}(n,k) \right) = \{X \in \mathbb{R}^{n\times k} : \|X\|_2 \leq 1\}$$
where $\|\cdot\|_2 $ is induced $2$-norm.
Question: Is there a characterization for the convex hull of the Stiefel manifold with  non-negativity constraints:
$$\mathrm{conv}(\mathrm{St}(n,k) \cap \mathbb{R}^{n\times k}_+)$$
where $\mathbb{R}^{n \times k}_+$ is the set of all $n \times k$ matrices with non-negative elements?

REPLY [2 votes]: @Arian has supplied the solution for $k = 1$.  It is a known result that for $k = n$, the desired convex hull is the set of all $n$ by $n$ doubly stochastic matrices.
I offer this conjecture for the solution for general $k$, as the following extension of the answer by @Arian .The remainder of this post is conjecture:
$$\mathrm{conv}(\mathrm{St}(n,k) \cap \mathbb{R}^{n\times k}_+) = \{X\in\mathbb{R}^{n\times k}_+:\|X\|_2 \le 1, \Sigma_{j=1}^n X_{ij} \ge 1\ \forall i\}$$
$\{X\in\mathbb{R}^{n\times k}_+:\|X\|_2 \le 1, \Sigma_{j=1}^n X_{ij} \ge 1\ \forall i\}$ implies $\|X\|_2 \ge \sqrt{\frac{k}{n}}$ and  $\sqrt{n-k+1} \ge \Sigma_{j=1}^n X_{ij}, \forall i$. 
So putting it altogether, we can state
$\mathrm{conv}(\mathrm{St}(n,k) \cap \mathbb{R}^{n\times k}_+) =$ $$\{X\in\mathbb{R}^{n\times k}_+: \sqrt{\frac{k}{n}} \le \|X\|_2 \le 1, \sqrt{n-k+1}  \ge \Sigma_{j=1}^n X_{ij} \ge 1\ \forall i\}$$
When $k = n$, $\|X\|_2 = 1, \Sigma_{j=1}^n X_{ij} = 1\ \forall i$ must hold, and this reduces exactly to the set of all $n$ by $n$ doubly stochastic matrices, thereby matching the known result for square $X$.