A codimension-one affine subspace of an affine space or a codimension-one linear subspace of a vector space; in projective geometry a hyperplane is a codimension-one projective subspace. In the affine case it is defined by a single linear equation and separates the ambient space into two half-spaces.

Principle

A hyperplane is given by the vanishing of one nontrivial affine (or homogeneous linear, in the projective case) functional; it is the simplest nontrivial flat (linear/affine) object and often serves as a supporting or separating structure in linear and convex geometry.

Demonstration

In R^3 the set {x in R^3 : ax + by + cz + d = 0} is an affine hyperplane (a plane) when (a,b,c) ≠ (0,0,0); in the projective plane a line is a projective hyperplane, described by a homogeneous linear equation.

Misapplication

Calling a curved two-dimensional surface in R^3 a hyperplane, or labeling a subspace of codimension greater than one as a hyperplane; forgetting the distinction between affine and projective contexts (affine hyperplanes separate, projective hyperplanes do not).

Consequence

Hyperplanes partition affine space into two half-spaces and are fundamental in linear algebra (kernels of linear functionals), convexity (supporting hyperplanes), classification (linear separators), and duality constructions in projective geometry.

Reversal

Replacing a hyperplane by a hypersurface allows nonlinear curvature and more complex local geometry; considering lower-codimension subspaces (lines, planes in higher dimension) reduces the separating power, while higher-codimension subspaces fail to separate.

Boundary

Applies to linear, affine, or projective settings and presupposes a vector or affine structure; excludes nonlinear hypersurfaces, manifolds without linear structure, and discrete sets; in projective space there is no separation into half-spaces.

Semantic Tension

Tension with the broader term 'hypersurface': hyperplanes are linear/homogeneous hypersurfaces while hypersurfaces may be nonlinear; tension also with the notion of 'hyperplane at infinity' which distinguishes affine from projective pictures.

Synthesis

A hyperplane is a flat codimension-one linear or affine subspace (or the projective analogue) defined by a single nontrivial linear equation; it is the basic linear separator and supporting object in algebraic, affine and projective geometry.