第二章 凸集 Convex Sets

2.1 仿射集与凸集 Affine and Convex Sets

2.1.1 线与线段 Lines and line segments

Suppose \(x_1\neq x_2\) are two points in \(\mathbb{R}^n\). Points of the form \[y = \theta x_1 + \left(1-\theta\right)x_2.\] where \(\theta \in \mathbb{R}\), form the line passing through \(x_1\) and \(x_2\). The parameter value \(\theta = 0\) corresponds to \(y = x_2\), and the parameter value \(\theta = 1\) corresponds to \(y = x_1\). Values of the parameter \(\theta\) between \(0\) and \(1\) correspond to the (closed) line segment between \(x_1\) and \(x_2\).

Expressing \(y\) in the form \[y = x_2 + \theta \left(x_1-x_2\right)\]