Half-space (geometry)

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. If the space is two-dimensional, then a half-space is called a half-plane (open or closed). A half-space in a one-dimensional space is called a half-line or ray.

More generally, a half-space is either of the two parts into which a hyperplane divides an n-dimensional space. That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane.

A half-space can be either open or closed. An open half-space is either of the two open sets produced by the subtraction of a hyperplane from the affine space. A closed half-space is the union of an open half-space and the hyperplane that defines it.

The open (closed) upper half-space is the half-space of all $x_{1},x_{2},\dots ,x_{n}$ such that $x_{n}\geq 0$ ($x_{n}>0$). The open (closed) lower half-space is defined similarly, by requiring that $x_{n}$ be negative (non-positive).

A half-space may be specified by a linear inequality, derived from the linear equation that specifies the defining hyperplane. A strict linear inequality specifies an open half-space: $$a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}>b.$$ A non-strict one specifies a closed half-space: $$a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\geq b.$$ Here, one assumes that not all of the real numbers a₁, a₂, ..., a_n are zero.

A half-space is a convex set.

See also

• Hemisphere (geometry)
• Line (geometry)
• Nef polygon, construction of polyhedra using half-spaces
• Poincaré half-plane model
• Quadrant (solid geometry)
• Siegel upper half-space

External links

• "Half-plane", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Half-Space". MathWorld.