I have a set of points $S\subset\mathbb{R}_+^d$, which always includes the origin. I am interested in the set of all points $x$ that can be written as the convex combination of $x_1\in S$ and $x_2 \in S$, where $x_2 \geq x_1$ elementwise.
Is there a name for the set of all such points? Or is this set perhaps a union of well-understood objects? I'm studying objects of this type, and am looking for keywords to search in the literature.
The linked figure shows an example of constructing such a set from a set of points (black circles). The shaded region is the set of interest. Example construction
