In my analysis textbook, I have the following definition of a convex set:
We call a set $E\subset\mathbb{R^n}$ convex if $$\lambda \vec{x}+(1-\lambda)\vec{y}\in\mathbb{E}$$ whenever $\vec{x},\vec{y}\in E$, and $0<\lambda<1$.
I then recalled seeing this in an explanation of Affine spaces:
Suppose two people, Alice and Bob, believe that two different points are the origin. Alice knows the actual origin, whilst Bob believes a point $p$ is the origin. For Bob, two vectors $\vec{a}$ and $\vec{b}$ will actually be $\vec{a}-\vec{p}$ and $\vec{b}-\vec{p}$, respectively. Then, clearly the sum $\vec{a}+\vec{b}$ will yeild a different solution for certain choices of $\vec{a}$ and $\vec{b}$. However, Alice and Bob will end up in the exact same place if Alice computes $$\lambda\vec{a}+(1-\lambda)\vec{b}$$ and Bob similarly computes $$\vec{p}+\lambda(\vec{a}-\vec{p})+(1-\lambda)(\vec{b}-\vec{p})$$
I'm really well versed in Affine spaces, but I know the basic intuition behind how they operate. My question is: why are affine combinations significant in both of these definitions, are they related in some way, or is this just a coincidence?
*Above explanation of affine space is from here.
The difference is that $\lambda$ ranges over $\mathbb{R}$ for affine spaces, while for convex sets $\lambda$ ranges over the interval $(0,1)$. So for any two points in a convex set $C$, the line segment between those two points is also in $C$. On the other hand, for any two points in an affine space $A$, the entire line through those two points is also in $A$.
So convex combinations are just a special case of affine combinations. A convex set turns into an affine space if you remove the restriction $0 < \lambda < 1$ from the definition.