Affine set
1 Line
Suppose x_1, x_2 are two points in \mathbb{R^n}. Then the line passing through them is defined as follows:
x = \theta x_1 + (1 - \theta)x_2, \theta \in \mathbb{R}
2 Affine set
The set A is called affine if for any x_1, x_2 from A the line passing through them also lies in A, i.e.
\forall \theta \in \mathbb{R}, \forall x_1, x_2 \in A: \theta x_1 + (1- \theta) x_2 \in A
3 Related definitions
3.1 Affine combination
Let we have x_1, x_2, \ldots, x_k \in S, then the point \theta_1 x_1 + \theta_2 x_2 + \ldots + \theta_k x_k is called affine combination of x_1, x_2, \ldots, x_k if \sum\limits_{i=1}^k\theta_i = 1.
3.2 Affine hull
The set of all affine combinations of points in set S is called the affine hull of S:
\mathbf{aff}(S) = \left\{ \sum\limits_{i=1}^k\theta_i x_i \mid x_i \in S, \sum\limits_{i=1}^k\theta_i = 1\right\}
3.3 Interior
The interior of the set S is defined as the following set:
\mathbf{int} (S) = \{\mathbf{x} \in S \mid \exists \varepsilon > 0, \; B(\mathbf{x}, \varepsilon) \subset S\}
where B(\mathbf{x}, \varepsilon) = \mathbf{x} + \varepsilon B is the ball centered at point \mathbf{x} with radius \varepsilon.
3.4 Relative Interior
The relative interior of the set S is defined as the following set:
\mathbf{relint} (S) = \{\mathbf{x} \in S \mid \exists \varepsilon > 0, \; B(\mathbf{x}, \varepsilon) \cap \mathbf{aff} (S) \subseteq S\}
