Given a point-to-set map $C: X \rightrightarrows Y$ defined by some vector valued-function $\mathbf{g}: X \times Y \to \mathbb{R}^n$ such that $C(x) \doteq \{y \in Y | g_1(x,y), …, g_n(x,y) \geq 0 \}$, I would like to impose conditions on $\mathbf{g}$ and $Y$ that guarantee the point-to-set map to be concave, i.e., for all $\lambda \in (0,1)$ and $x, x^\prime \in X$, $C(\lambda x + (1-\lambda) x^\prime) \subseteq \lambda C(x) + (1-\lambda) C(x^\prime)$ where the sum is defined as the sum and scalar multplication are minkowski (the definition of concave point to set map can be found here).
I have seen in some papers (such as this one, see examples 2.3-2.5) certain examples of $\mathbf{g}$ that guarantee concavity of C, but these conditions often require special structures on $\mathbf{g}$, such as $\mathbf{g}$ being scalar-valued I was wondering if there are more general conditions I can impose on $g$ that are more general that guarantee C to be concave? The one condition that I thought would guarantee concavity has been that g is affine in (x,y), however, I have run into counter examples.