Half-space

A half-space is either of the two regions into which a hyperplane divides a real Euclidean space, with or without the dividing hyperplane itself. Including the boundary gives a closed half-space; excluding it gives an open half-space. Half-spaces provide the geometric interpretation of linear inequalities and are basic building blocks of convex sets. (web.stanford.edu)

Definition and geometry

For a nonzero vector $a\in\mathbb R^n$ and a scalar $b\in\mathbb R$, a closed half-space is

Its boundary is the hyperplane

Here $a^{\mathsf T}x$ is the standard inner product. The vector $a$ is perpendicular to $P$ and points outward from $H_{\leq}$, toward increasing values of $a^{\mathsf T}x$. Reversing the inequality selects the opposite side. (web.stanford.edu)

Thus, “half” identifies a side of a dividing hyperplane, not a finite region of half the space’s volume. In $\mathbb R^2$, the region is a half-plane; in $\mathbb R^3$, it is bounded by an ordinary plane. Multiplying both $a$ and $b$ by a positive scalar leaves the defining inequality unchanged. (web.stanford.edu)

Open and closed half-spaces

The strict inequality

defines an open set, whereas $H_{\leq}$ is a closed set. Their boundary is $P$; the interior of $H_{\leq}$ is $H_<$, and the closure of $H_<$ is $H_{\leq}$. These descriptions also hold in a real Hilbert space when $a^{\mathsf T}x$ is replaced by $\langle a,x\rangle$. (arxiv.org)

The two opposite open half-spaces are disjoint. The two opposite closed half-spaces intersect exactly in their boundary hyperplane. Consequently,

(stanford.edu)

Convexity

Every half-space is convex. For example, if $x,y\in H_{\leq}$ and $0\leq t\leq1$, then

Hence every convex combination of two points in the half-space remains inside it. The same argument applies to strict inequalities. (courses.csail.mit.edu)

Intersections and optimization

A polyhedron is an intersection of finitely many closed half-spaces. Using a matrix $A$, it can be written

where the inequalities are interpreted componentwise. Equality constraints can be represented by pairs of opposite inequalities. Such intersections may be bounded, unbounded, lower-dimensional, or empty; they are convex because intersections preserve convexity. (stanford.edu)

In linear programming, these intersections describe the feasible set. Half-spaces also express separation: a hyperplane can place a closed convex set on one side and an exterior point strictly on the other. This connects linear inequalities with geometric certificates of infeasibility. (courses.csail.mit.edu)

Distance and projection

For $H_{\leq}=\{x:a^{\mathsf T}x\leq b\}$, the distance from a point $z$ to the half-space is

Its nearest-point projection is

A point already inside is unchanged; an exterior point moves perpendicularly to the boundary. The formula extends to real Hilbert spaces and supports projection algorithms for systems of inequalities. (arxiv.org)

Nondegeneracy

The requirement $a\ne0$ is essential. If $a=0$, the inequality $a^{\mathsf T}x\leq b$ describes the whole space when $b\geq0$, and the empty set when $b<0$; neither has a dividing hyperplane. A half-space should also be distinguished from its boundary: the half-space is full-dimensional, while the hyperplane has dimension $n-1$. (web.stanford.edu)

References

1. Convex Optimizationstanford.edu
2. Convex Optimization — Lecture Slidesweb.stanford.edu
3. Linear Programming Lecture Notescourses.csail.mit.edu
4. Projecting onto intersections of halfspaces and hyperplanesarxiv.org