The difference between affine set and affine hull

Question

According to the definition of affine hull and affine set.
$$aff [C] = [\theta_1x_1+...+\theta_nx_n|x_1,...x_n \in C, \theta_1+...+\theta_n=1] $$
The data in affine hull is also in affine set. And vice versus. So what's difference between them?

Answer 1

Affine set, Affine hull, Convex set and Convex hull

Affine set is a set which contains every affine combinations of points in it. For example, for two points $x,y\in R^2 $, an affine set is the whole line passing through these two points. (Note: $\theta_i$ could be negative as long as $\theta_1 + \theta_2 =1$. If all $\theta_i \geq 0$, it is called a convex set and it is the line segment between two points instead of the whole line.)

For affine hull, convex hull might be easier to visualize first. Let's say $A= \{ x,y,z | x,y,z \in R^2 \} $. Convex hull of A is a plastic wrap around x,y,z containing every points within the wrap and on the border. The reason it is within the wrap is because coefficients $\theta_i \geq 0 $. affine hull allow $ \theta_i $ to be negative, so affine hull will be entire $R^2$ unless all three points lie on a line, in that case , affine hull will be the whole line passing through these three points.

Defintion of Affine hull for an arbitrary set A is the smallest affine set that can contains all points in A. Affine hull for an affine set is the affine set itself.

Hope it helps !!

Answer 2

$A$ is an affine set iff for any $x,y \in A$ the line passing through $A$ is also contained in $A$.

The affine hull of a set $S$, denoted by $\operatorname{aff} S$, is defined to be the smallest affine set that contains $S$. (By smallest, I mean the intersection of all affine sets containing $S$.)

At some level the terms mean the same thing in the sense that an affine hull is an affine set, and any affine set can be written as the affine hull of itself, but generally when the term affine hull is used, the underlying constituent set is being highlighted in some way.

Answer 3

aff C is the smallest affine set that contains set C. So by definition a affine hull is always a affine set.

e.g.

• The affine hull of 3 points in a 3-dimensional space is the plane passing through them.
• The affine hull of 4 points in a 3-dimensional space that are not on the same plane is the entire space.