# Halfspace

Consider the hyperplane, $LaTeX: \{x \in \mathbb{R}^n: a^T x = b\}$ (where $LaTeX: a \in \mathbb{R}^n \backslash {0}$). This partitions $LaTeX: \mathbb{R}^n$ into two (closed) half spaces: $LaTeX: \{ x: a^Tx \le b\}$ and $LaTeX: \{ x: a^Tx \ge b\}$. The two associated open halfspaces are $LaTeX: \{ x: a^t x < b\}$ and $LaTeX: \{ x: a^Tx > b\}$.