Convex Set or Convex Space?

Solution 1:

The typical example of a set with linear properties is $\mathbb{R}^n$ or $\mathbb{C}^n$. These look like real space. For this reason, the word "space" is nicely indicative of how linearly closed sets look, act, and feel like in general. They're also enough to be a full "space" that an operation works on. In particular, linearly closed subspaces of vector spaces are vector spaces themselves.

Convex sets have a lot more variety to the shapes they can look like. They can be circles, ovals, things like $[0,1] \times \mathbb{R}$, and don't have a clear analogy to "space" in a real-world sense. Also, convex subsets of vector spaces can fail to be subspaces. So it would be a bit misleading to use the word "space" to describe them.

Solution 2:

Usually when people say space they mean a linear space. That is whenever the points $A,B$ are in the space then the line $tA+(1-t)B$ is completely in the space for all $t$ in $(-\infty,\infty)$.

That's why we usually refer to convex sets because we only require whenever the points $A,B$ are in the set then the line segment $tA+(1-t)B$ is completely in the space for all $t$ in $[0,1]$.