Consider the Stiefel manifold

$$\mathrm{St}(n,k) :=\{X \in \mathbb{R}^{n\times k} : X^TX = I_k\},$$

where, $I_k$ is $k$ dimensional identity matrix. It is well known that

$$\mathrm{conv}(\mathrm{St}(n,k))= \{X \in \mathbb{R}^{n\times k} : \|X\|_2 \leq 1\},$$

where, $\|\cdot\|_2 $ is induced 2-norm of an operator.

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?

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?