Half-Space — Definition, Formula & Examples

A half-space is one of the two regions into which a plane (in 3D) or a line (in 2D) divides the entire space. Every linear inequality, such as

2x + 3y \leq 6

, defines a half-space containing all points that satisfy that inequality.

Given a hyperplane $H = \{\mathbf{x} \in \mathbb{R}^n : \mathbf{a} \cdot \mathbf{x} = b\}$ where $\mathbf{a} \neq \mathbf{0}$ , the two associated closed half-spaces are $H^+ = \{\mathbf{x} \in \mathbb{R}^n : \mathbf{a} \cdot \mathbf{x} \geq b\}$ and $H^- = \{\mathbf{x} \in \mathbb{R}^n : \mathbf{a} \cdot \mathbf{x} \leq b\}$ . An open half-space uses strict inequality.

Key Formula

H^- = \{\mathbf{x} \in \mathbb{R}^n : \mathbf{a} \cdot \mathbf{x} \leq b\}

Where:

$\mathbf{x}$ = A point in n-dimensional space
$\mathbf{a}$ = The normal vector to the boundary hyperplane (nonzero)
$b$ = A real constant defining the offset of the hyperplane

How It Works

To determine which half-space a point belongs to, substitute its coordinates into the linear expression

\mathbf{a} \cdot \mathbf{x}

and compare the result to

b

. If the result is less than

b

, the point lies in

H^-

; if greater, it lies in

H^+

; if equal, the point lies on the boundary hyperplane itself. In linear programming, each constraint defines a half-space, and the feasible region is the intersection of all such half-spaces — a convex polytope.

Worked Example

Problem: Determine whether the point (1, 4) lies in the half-space defined by 2x + 3y ≤ 12.

Substitute: Plug the coordinates into the left side of the inequality.

2(1) + 3(4) = 2 + 12 = 14

Compare: Check whether the result satisfies the inequality

\leq 12

14 \leq 12 \quad \text{is false}

Answer: The point (1, 4) does not lie in the half-space

2x + 3y \leq 12

. It lies on the opposite side of the boundary line.

Why It Matters

Every constraint in a linear programming problem defines a half-space, so understanding half-spaces is essential for courses in optimization, operations research, and convex analysis. Support vector machines in machine learning also rely on finding an optimal hyperplane that separates data into two half-spaces.

Common Mistakes

Mistake: Confusing a closed half-space (≤ or ≥) with an open half-space (< or >).

Correction: A closed half-space includes the boundary hyperplane as part of the region; an open half-space excludes it. In linear programming, constraints are typically closed, meaning boundary points are feasible.