Half space

In mathematics, a half-space is a subset of a space of any dimension, bounded by a hyperplane . If the hyperplane itself is contained in the half-space, this is called closed , otherwise open . The term half-space is derived from the fact that the delimiting hyperplane divides the space into two parts. Terminology and imagination are a generalization from three-dimensional visual space, where a plane delimits a half-space.

Formal definition

Special case ℝ ⁿ

For , and the standard scalar product is called ${\displaystyle a\in \mathbb {R} ^{n}\setminus \{0\}}$ ${\displaystyle \beta \in \mathbb {R} }$ ${\displaystyle \langle \cdot ,\cdot \rangle }$

${\displaystyle \{x\in \mathbb {R} ^{n}\mid \langle a,x\rangle =\beta \}}$

a hyperplane ,

${\displaystyle \{x\in \mathbb {R} ^{n}\mid \langle a,x\rangle \geq \beta \}}$

a closed half-space and

${\displaystyle \{x\in \mathbb {R} ^{n}\mid \langle a,x\rangle >\beta \}}$

an open half-space .

general definition

Let it be a real vector space . Then for each linear form with and each is called the subset ${\displaystyle V}$ ${\displaystyle \lambda \colon V\to \mathbb {R} }$ ${\displaystyle \lambda \neq 0}$ ${\displaystyle \beta \in \mathbb {R} }$

${\displaystyle \{v\in V\mid \lambda (v)\geq \beta \}}$ or. ${\displaystyle \{v\in V\mid \lambda (v)>\beta \}}$

a closed or open half-space .

Affine spaces

The general definition for real vector spaces of any dimension can be transferred to finite-dimensional affine spaces over an ordered body . The transferred term is generalized in synthetic geometry in the two-dimensional case to affine incidence levels. → See also the division of pages .

Illustrative special cases

• On a straight line , the hyperplanes are exactly the points , and a half-space is thus a subset of the straight line delimited by a point . In this special case one also speaks of a half-line . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {R} }$
• In the plane , the hyperplanes are exactly the straight lines , and thus a half-space is a subset of the delimited by a straight line . In this special case one speaks of a half plane . ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{2}}$
• The hyperplanes of space are precisely the planes , and a half-space is a three-dimensional subset of space bounded by a plane. ${\displaystyle \mathbb {R} ^{3}}$