In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space.

In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap.

The hyperplane separation theorem is due to Hermann Minkowski.

The Hahn–Banach separation theorem generalizes the result to topological vector spaces.

A related result is the supporting hyperplane theorem.

If both sets are closed, and at least one of them is compact, then the separation can be strict, that is,

The summary of the results are as follows: The number of dimensions must be finite.

In infinite-dimensional spaces there are examples of two closed, convex, disjoint sets which cannot be separated by a closed hyperplane (a hyperplane where a continuous linear functional equals some constant) even in the weak sense where the inequalities are not strict.

[5] Here, the compactness in the hypothesis cannot be relaxed; see an example in the section Counterexamples and uniqueness.

Since the distance function is continuous, there exist points

in fact have the minimum distance over all pairs of points in

is closed and compact, and the unions are the relative interiors

Since the unit sphere is compact, we can take a convergent subsequence, so that

If both sets are open, then there exist a nonzero vector

have possible intersections, but their relative interiors are disjoint, then the proof of the first case still applies with no change, thus yielding: Separation theorem II — Let

, then extend the affine span to a supporting hyperplane.

Note that the existence of a hyperplane that only "separates" two convex sets in the weak sense of both inequalities being non-strict obviously does not imply that the two sets are disjoint.

For example, if A is a closed half plane and B is bounded by one arm of a hyperbola, then there is no strictly separating hyperplane: (Although, by an instance of the second theorem, there is a hyperplane that separates their interiors.)

In the first version of the theorem, evidently the separating hyperplane is never unique.

Technically a separating axis is never unique because it can be translated; in the second version of the theorem, a separating axis can be unique up to translation.

The horn angle provides a good counterexample to many hyperplane separations.

, the unit disk is disjoint from the open interval

is relatively open, then there does not necessarily exist a separation that is strict for

[6] Farkas' lemma and related results can be understood as hyperplane separation theorems when the convex bodies are defined by finitely many linear inequalities.

[6] In collision detection, the hyperplane separation theorem is usually used in the following form: Separating axis theorem — Two closed convex objects are disjoint if there exists a line ("separating axis") onto which the two objects' projections are disjoint.

The separating axis theorem can be applied for fast collision detection between polygon meshes.

Each face's normal or other feature direction is used as a separating axis.

In 3D, using face normals alone will fail to separate some edge-on-edge non-colliding cases.

Additional axes, consisting of the cross-products of pairs of edges, one taken from each object, are required.

[7] For increased efficiency, parallel axes may be calculated as a single axis.