Convex join

Remarks

Example: in $\R^2$, let $A = \set{(0, 0)}$ and $B = [(1, 0), (1, 1)]$. The join is the union of the segments from the origin to the points $(1, s)$ with $0 \le s \le 1$, which is the closed triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$.

If $A$ and $B$ are nonempty convex sets, then $\operatorname{join}(A, B)$ equals the convex hull of $A \cup B$. Indeed, a convex combination of points of $A \cup B$ can be regrouped as $(1 - t) a + t b$ with $a \in A$ and $b \in B$.

Non-example of that identity: if $A = \emptyset$ and $B = \set{b}$, then $\operatorname{join}(A, B) = \emptyset$, while the convex hull of $A \cup B$ is $\set{b}$. So the nonemptiness hypothesis cannot be dropped.