angle constrained convex hull

Question

Given a set of points $P$ in $R^d$ it is straightforward to compute the convex hull (Graham-scan etc). However, the angle between the adjacent faces are unconstrained. Let us suppose two adjacent faces with normal $\vec{n}_1$ and $\vec{n}_2$. Let us suppose we want $\vec{n}_1.\vec{n}_1 \leq \cos \alpha$. (See figure for illustration) How do we generate such a convex hull ? Which algorithm (with suitable modifications) do I use ?

Constrained convex hull. The image shows the convex hull of a set of points. Further, a violation on angular constraint is marked. A required conv hull is shown in dashed