Convex Set or Convex Space? Is there a reason why we usually refer to sets with linear, or affine properties, as spaces, and to sets with convex properties as sets ? Shouldn't we call them convex spaces instead of convex sets ?
Obs: I have seen affine sets and affine spaces being used interchangeably, but not linear spaces, nor convex sets.
 A: 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.
A: 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 $[0,1]$. 
A: I don't have much authority to answer here, but this is my best guess.
Both affine and linear spaces inherit axiom structures. If you talk about a linear subspace, it is, itself, a linear space in its own right. Similarly, if you take an affine subspace of an affine space, it is an affine subspace in its own right. As it happens, you can also take an affine subspace of a linear space in a natural way, which is a weaker notion, but still falls into an axiomatic framework.
A convex set does not. It has its own algebraic structure, specifically, a line segment between two points must be contained in the set, but such a structure relies on the affine structure of the space it's contained in. It's not so easy to consider convex sets as algebraic spaces in their own right; I've certainly never seen axioms for them. It's difficult to describe a line segment without affine structure, and it's even harder to talk about being "in the set" when you disregard everything out of the set!
My guess is, of course, a little spurious at best, since vector spaces rely on ground fields, and affine spaces rely on an entire vector space. But, I do think that, in some way, convexity is even less independent than linear or affine spaces.
It's also worth comparing to topology (which I'm guessing you might have picked up this enthusiasm for the word "space"). In topology, every subset can be naturally turned into a topological space in its own right. This is just not true in linear algebra.
A: This sounds like you got confused by set theoristic terminology. In type theory, types can be equipped with a convex space structure, and convexity of sets is defined for sets of elements of one such type. Note that I use here "convex space" to mean "affine space without needing subtraction".
I invite you to read the formal definition of convex sets in mathlib (we don't have convex spaces yet). Hopefully, this will clear things up.
A: From Wikipedia, a mathematical space is a set with some added structure. Typical structures include measures (norm, distance), algebraic structures, topologies, etc.. Although there is no formal definition of what constitutes a space. Also, refer to 3 for a discussion of the difference between a space and a structure.
Convexity doesn't introduce a new structure in the sense of 2. It is just a special type of subsets of a vector space. It merely adds a restriction that any line segment between two points within the set should be contained within the set itself. One could say that this is indeed a geometrical structure but this structure is not as powerful as the types of structures considered in 2. The addition of two vectors gives another vector (in a vector space). Union of two open sets gives another open set (in topological or metric spaces). But no such operation is available for convex sets.
There are many other special types of subsets of a vector space that are not spaces. Cones (not necessarily convex), half-spaces, spheres (not ball), ellipsoids, simplexes, polyhedrons, are few examples. In fact, normally, a special class/type of sets is just a set with some useful properties. Only in rare cases, the properties are so useful that it gets elevated to the notion of a mathematical space.
BTW, there is something called Locally convex topological vector space (TVS).
