Separating HyperplaneIt is a plane that splits two non-intersecting convex sets into two parts.

In mathematics, a hyperplane is a linear subspace in an n-dimensional Euclidean space with a codimension equal to 1. For low dimensions, it is a straight line in a plane or a plane in space.

Separating Hyperplane Theorem

If there are two union sets C and D (disjoint, i.e. C ∩ D = ∅), and both sets are convex,

Then there must exist a hyperplane (a hyperplane is both a convex set and an affine set),

So that for all points x in the set C, a ^T x ≤ b , x ∈ C, all points x in set D satisfy a ^T x ≥ b, x ∈ D,

In other words, the affine function a ^T – b is non-positive on set C and non-negative on set D.

Hyperplane { x | a ^T = b } is called the dividing hyperplane of sets C and D, as shown in the figure below.

Converse Theorem

Converse separating hyperplane theorems:

For any two convex sets C and D, at least one of which is open, then sets C and D are disjoint if and only if there exists a separating hyperplane between them.

Related words: affine set, convex optimization

Sub-word: Hyperplane