21. Hyperplane

A hyperplane in ${ℝ}^{𝑛}$ is an $\left(𝑛-1\right)$-dimensional affine subspace — the natural generalization of a line in ${ℝ}^2$ or a plane in ${ℝ}^3$.

Defined by a single linear equation:

where $\overrightarrow{𝑎}\ne \mathrm{𝟎}$ is the normal vector and $𝑏\in ℝ$ is the offset.

21.1. Dimension of a hyperplane

In ${ℝ}^{𝑛}$, the equation $\overrightarrow{𝑎}\cdot \overrightarrow{𝑥}=𝑏$ imposes one linear constraint. The solution set has dimension $𝑛-1$:

• ${ℝ}^2$: hyperplane = line
• ${ℝ}^3$: hyperplane = plane
• ${ℝ}^{𝑛}$: hyperplane = $\left(𝑛\{-\}1\right)$-dimensional flat

21.2. Through the origin

If $𝑏=0$, the hyperplane passes through the origin and is a linear subspace (closed under addition and scalar multiplication). Otherwise it’s an affine subspace — a translated subspace.

21.3. Distance from a point to a hyperplane

Given hyperplane $\overrightarrow{𝑎}\cdot \overrightarrow{𝑥}=𝑏$ and point $\overrightarrow{𝑝}\in {ℝ}^{𝑛}$:

(See Norm for $\Vert \overrightarrow{𝑎}\Vert$.)

21.4. Half-spaces

A hyperplane splits ${ℝ}^{𝑛}$ into two half-spaces:

These are the building blocks of polyhedra (intersections of finitely many half-spaces).

21.5. Where hyperplanes show up

• Linear programming: feasible regions are intersections of half-spaces (polyhedra)
• Support vector machines: the decision boundary is a hyperplane $\overrightarrow{𝑤}\cdot \overrightarrow{𝑥}+𝑏=0$
• Planes in ${ℝ}^3$ — the $𝑛=3$ case
• Linear constraints in optimization: each ${\overrightarrow{𝑎}}_{𝑖}^{𝑇}\overrightarrow{𝑥}\le {𝑏}_{𝑖}$ defines a half-space