geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, a half-space is either of the two parts into which a

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

divides the three-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

. If the space is

two-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...

, then a half-space is called a half-plane (open or closed). A half-space in a

one-dimensional In physics and mathematics, a sequence of ''n'' numbers can specify a location in ''n''-dimensional space. When , the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the ...

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

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...

. That is, the points that are not incident to the hyperplane are partitioned into two

convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...

s (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 set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

s 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. A half-space may be specified by a linear inequality, derived from the linear equation that specifies the defining hyperplane. A strict linear

inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...

specifies an open half-space: :

a_1x_1+a_2x_2+\cdots+a_nx_n>b

A non-strict one specifies a closed half-space: :

a_1x_1+a_2x_2+\cdots+a_nx_n\geq b

Here, one assumes that not all of the real numbers ''a''₁, ''a''₂, ..., ''a''_''n'' are zero.

Properties

* A half-space is a

Upper and lower half-spaces

The open (closed) upper half-space is the half-space of all (''x''₁, ''x''₂, ..., ''x''_''n'') such that ''x''_''n'' > 0 (≥ 0). The open (closed) lower half-space is defined similarly, by requiring that ''x''_''n'' be negative (non-positive).

External links

* * {{DEFAULTSORT:Half-Space Euclidean geometry

Properties

Upper and lower half-spaces

See also

External links