I'm considering the following minimization problem: $$ \begin{cases} min f(x) \\ s.t.\ \ \ \ g_{i}(x)\le0 & x \in R^n, i \in I={1,..m} & f,g_{i} \in C^1 \end{cases} \\ $$
Given $x \in S $ where S is the feasible region
$ Set: \\ D1(\bar x)=\{d \in R^n : \exists \alpha>0 : x+\alpha d \in S \}\\ I(\bar x)=\{i \in I: g_{i}(\bar x)=0 \} \\ D2(\bar x)=\{d \in R^n : \nabla^tg_{i}d \le0 \hspace{0.2 cm} \forall i \in I(\bar x) \} \\ $
We say that the qualification constraint (QC) holds at $\bar x$ if $ D1(\bar x)=D2(\bar x) $.
Now, according to my notes, if the $g_{i}$'s are convex and there exixsts a point $\alpha$ s.t. $g_{i}(\alpha)<0$ $ \forall i$ then the QC holds. I'm having some trouble with this last part. If a function is convex then the lower contours are convex sets so if, for istance, I take only one constraint, say: $g=x^2 + y^2-3, $ which is convex, then the feasible region is a disk and for each point of the circle the tangent t belongs to D2 but not to D1. So I'm thinking that maybe the $g_{i}$ must be concave or maybe I'm missing something.