Separating Hyperplane
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.