21. Hyperplane

A hyperplane in is an -dimensional affine subspace — the natural generalization of a line in or a plane in .

Defined by a single linear equation:

where is the normal vector and is the offset.

21.1. Dimension of a hyperplane

In , the equation imposes one linear constraint. The solution set has dimension :

• : hyperplane = line
• : hyperplane = plane
• : hyperplane = -dimensional flat

21.2. Through the origin

If , 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 and point :

(See Norm for .)

21.4. Half-spaces

A hyperplane splits 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
• Planes in — the case
• Linear constraints in optimization: each defines a half-space