Consider a set of points $S \subset \mathbb{R}^n$
Let $$H(S) = \\\{y \in \mathbb{R}^n, \forall a \in {\mathbb{R}^{+}}^n, \exists x \in S, \langle a,x-y\rangle\geq 0\}$$
This is the set of point for which are "below" at least one point in S under any linear projection with positive components.
In a same way the convex hull can be defined as the set of points which are less extremal than a point in S for any projection.
What can be said about this hull? Does it have a name? Known properties?