In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint.

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.

In the context of support-vector machines, the optimally separating hyperplane or maximum-margin hyperplane is a hyperplane which separates two sets of points and maximizes its distance to both sets.

Hyperplane separation theorem—Let ${\displaystyle A}$ and ${\displaystyle B}$ be two disjoint nonempty convex subsets of ${\displaystyle \mathbb {R} ^{n}}$. Then there exist a nonzero vector ${\displaystyle v}$ and a real number ${\displaystyle c}$ such that

for all ${\displaystyle x}$ in ${\displaystyle A}$ and ${\displaystyle y}$ in ${\displaystyle B}$; i.e., the hyperplane ${\displaystyle \langle \cdot ,v\rangle =c}$, ${\displaystyle v}$ the normal vector, separates ${\displaystyle A}$ and ${\displaystyle B}$.

If both sets are closed, and at least one of them is compact, then the separation can be strict, that is, ${\displaystyle \langle x,v\rangle >c_{1}\,{\text{ and }}\langle y,v\rangle <c_{2}}$ for some ${\displaystyle c_{1}>c_{2}}$

In all cases, assume ${\displaystyle A,B}$ to be disjoint, nonempty, and convex subsets of ${\displaystyle \mathbb {R} ^{n}}$. The summary of the results are as follows: