Supporting hyperplane theorem

The theorem asserts that if C is a nonempty closed convex subset of Rⁿ and x₀ is a boundary point of C, then there exists a nonzero vector a such that a·x ≤ a·x₀ for all x in C; the hyperplane {x : a·x = a·x₀} is then a supporting hyperplane.¹ A standard proof uses the separating hyperplane theorem: since x₀ is not in the interior, one can separate the singleton {x₀} from the interior of C, then pass to the limit to obtain a supporting hyperplane.² The theorem also holds in infinite-dimensional normed spaces under additional assumptions, such as when the set has nonempty interior or is locally convex.

In convex optimization, supporting hyperplanes provide necessary conditions for optimality: at a minimizer of a convex function over a convex set, the gradient (if it exists) defines a supporting hyperplane to the sublevel set.³ In geometry, the theorem implies that every convex polytope can be described as the intersection of half-spaces defined by its supporting hyperplanes, a fact used in computational geometry and linear programming.⁴ The concept also underlies the definition of the normal cone in convex analysis, which collects all normals of supporting hyperplanes at a point.

The theorem generalizes to infinite-dimensional Banach spaces via the Hahn–Banach theorem, but only for closed convex sets with nonempty interior or in reflexive spaces.⁵ In nonsmooth analysis, the notion of a Clarke tangent cone extends supporting hyperplanes to nonconvex sets, though the existence is not guaranteed. The theorem is closely related to the Minkowski–Farkas lemma and the separating hyperplane theorem, which are used in duality theory and game theory.⁶

For a convex set with empty interior (e.g., a line segment in R²), the theorem still holds, but the supporting hyperplane may not be unique; in fact, every boundary point of a convex set has a whole cone of supporting hyperplanes.¹ The theorem fails for nonconvex sets: a point on the boundary of a nonconvex set may have no supporting hyperplane, as seen in a crescent-shaped region. In infinite dimensions, the theorem requires the set to be closed and convex, but also that the point be a boundary point in the norm topology; otherwise, counterexamples exist using non-closed convex sets.⁵ The theorem was first proved by Hermann Minkowski in 1911 in the context of convex bodies, and it later became a cornerstone of convex geometry and functional analysis.

Convex set

A set where the line segment between any two points lies entirely within the set.

Boundary point

A point that is in the closure of the set but not in its interior.

Hyperplane

A set of the form {x : a·x = b} for some nonzero vector a and scalar b.

Supporting hyperplane

A hyperplane that touches a set at a boundary point and leaves the set entirely on one side.