Half-space (geometry)

In mathematics, a half-space is limited by a hyperplane subset of a space of arbitrary dimension. If the hyperplane is itself contained in the half space, the name of this completed, otherwise open. The term half-space is derived from the fact that the limiting hyperplane divides the space into two parts. Terminology and presentation are a generalization of the three-dimensional space of intuition, where a level defines a half-space.

Formal definition

Special case

For and called

A hyperplane

A closed half- space and

An open half-space.

General definition

It is a real vector space. Then for every linear form and each subset is called

A closed or open half-space.

Affine spaces

The general definition for real vector spaces of arbitrary dimension can be applied to finite-dimensional affine spaces over a subordinate body. The transmitted concept is generalized in synthetic geometry in two-dimensional case to affine incidence levels. → See page classification.

Illustrative special cases

• In a straight line the hyperplanes are precisely the points, and a half-space is thus a point delineated by a subset of the line. In this special case, one also speaks of a half-line.
• In the plane of the hyperplanes are exactly the straight line, and thus a half-space is a straight line defined by a subset of the. In this special case, one also speaks of a half-plane.
• The hyperplanes of the space are exactly the plains, and a half-space is limited by a plane three-dimensional subset of the space.

• Geometry