I haven't been able to find a clear answer to this question, seek an answer from a professor or figure it out myself as I am not a mathematics expert. I used the nonlinear-optimization but I'm not sure this is specific enough, so I invite you to add more correct tags.
I have a specific problem, but I'll try to phrase it more generally: What are convex constraints? Here's the frame of my problem.
Consider a non-convex quadratic optimization problem, with the following cost function, where X is given:
$$\|\textbf{X} - \textbf{XCS}\|_F^2$$
subject to constraints $|\textbf{c}_j|_1 = 1\,\, \forall \,\, j \in \{0, ..., k\}$ and $|\textbf{s}_i|_1 = 1 \,\, \forall \,\, i \in \{0, ..., n\}$ and "all coefficients in $\textbf{C}$ and $\textbf{S}$ are positive". Furthermore, $\textbf{X} = [\textbf{x}_1, \textbf{x}_2, ..., \textbf{x}_n] \in \mathbb{R}^{m \times n}$, $\textbf{S} \in \mathbb{R}^{k \times n}$ and $\textbf{C} \in \mathbb{R}^{n \times k}$.
I can show that my problem is non-convex by showing that the Hessian is not positive semi-definite, but how do I show that my constraints are convex? What does it mean to the optimization problem that they are. What does it mean that they aren't?