What is affine hull of conv(A)

Question

Consider the set $A = \{(1,0),(0,1),(-1,0),(0,-1)\}$.
The convex hull of $A$, i.e. $conv(A)$, should look like the following:
(This is also a $l_1$-norm unit ball.)

My question is what is the affine hull of $conv(A)$?

My answer is the whole $R^2$-space, i.e. the whole plane. Is it true?
Can anyone give me more information about affine hull of $conv(A)$?

Thanks

Answer 1

You can simply solve some linear equations to show that any point in the plane can be expressed as an affine combination of three vertices, for example $(\pm 1,0)$ and $(0,1)$.

$$ \begin{aligned} (x,y)\ &= a(1,0) + b(-1,0) +c(0,1)\\ 1\ &= a+b+c \end{aligned} $$

$$ \Rightarrow (x,y) = \frac{1-y+x}{2}(1,0) + \frac{1-y-x}{2}(-1,0) + y(0,1) $$

Hence the affine hull of $A$ is the entire plane.