Affine sets

A set $C\subseteq\mathbb R^n$ is affine if the line through any two points in $C$ lies in $C$. Namely, for any $x_1, x_2\in C$ and $\lambda\in\mathbb R$, we have $\lambda x_1+(1-\lambda)x_2\in C$

If $C$ is an affine set then $C$ can be expressed as

$$C = x_0 + V=\{x_0+v|\ v\in V\},$$

where $x_0$ is a point in $C$ and $V$ is a subspace. It means that for any affine set $C$, there is a subspace associated with $C$. We define the dimension of $C$ as the simension of the subspace $V$.

Given a set $C$ and $x_1,\ldots,x_n\in C$, we refer to a point of form $\lambda_1x_1 + \ldots + \lambda_n x_n$, where $\lambda_1+\ldots + \lambda_n=1$, as an affine combination of the points $x_1,\ldots,x_n$.

The set of all affine combinations of points in $C$ is called the affine hull of $C$, i.e.

$$\text{aff}(C) = \left\{ \sum_{i=1}^n \lambda_i x_i\ \Big|\ \ x_i\in C, \lambda_i\in\mathbb R \text{ and} \sum_{i=1}^n\lambda_i=1 \right\}.$$

Note: The affine hull of $C$ is the smallest affine set that contains $C$.

Convex sets

A set $C$ is convex if the line segment between any two points in $C$ lies in $C$. Namely, for any $x_1, x_2\in C$ and $\lambda\in\mathbb [0,1]$ we have $\lambda x_1+(1-\lambda)x_2\in C$

Note: Different from the definition of affine sets, in a convex set we only need the segment lies on it, which characterized by the range of $\lambda$

Roughly speaking, in a convex set, any point can be seen by each other through an straight path between them. Every affine set is also convex.

Given a set $C$ and $x_1,\ldots,x_n\in C$, we call to a point of form $\lambda_1x_1 + \ldots + \lambda_n x_n$, where $\lambda_i\geqslant0$ and $\lambda_1+\ldots + \lambda_n=1$, a convex combination of the points $x_1,\ldots,x_n$.

The set of all convex combination of points in $C$ is called the convex hull of $C$

$$\text{conv}(C) = \left\{ \sum_{i=1}^n \lambda_i x_i\ \Big|\ \ x_i\in C, \lambda_i\geqslant 0 \text{ and} \sum_{i=1}^n\lambda_i=1 \right\}.$$

Note: In the definition of affine hull, $\lambda_i\in\mathbb R$, while in the definition of convex hull, $\lambda_i\geqslant0$