The ordinary convex hull $\text{conv} A$ of a set $A$ in Euclidean space may be defined as the intersection of all convex supersets of $A$. Suppose I wish to restrict the permissible supersets as follows:
Definition. Let $A,D\subset\mathbb{R^n}$. The D-directional convex hull $\text{conv}_D A$ is the intersection of all convex supersets of $A$ with an outer normal in $D\setminus\{0\}$ at every point on their boundary. (By the outer normal I mean the normal of a supporting hyperplane.)
Has this concept been studied before? If so, under which name?
It is different from the orientation-restricted convex hull.