For a nonempty convex set K in a real vector space with an inner product, the support function h_K(u) assigns to each direction u the supremal inner product sup_{x∈K}⟨x,u⟩; equivalently it records the signed distance of the supporting hyperplane normal to u.

Principle

h_K is positively homogeneous of degree one and convex as a function of direction; it linearizes Minkowski addition via h_{K+L}=h_K+h_L and encodes boundary support information. Translations change h_K by a linear term: h_{K+t}(u)=h_K(u)+⟨t,u⟩.

Demonstration

The support function of the Euclidean ball of radius r is h(u)=r||u||; for a polytope it equals the maximum of finitely many linear functionals determined by its vertices or facets. The support function uniquely determines a compact convex body up to the position encoded by linear terms.

Misapplication

Using h_K as if it were the radial function (distance from origin to boundary along u) without accounting for translation effects, or applying convexity-based conclusions when K is nonconvex; failing to track the linear displacement term after translation.

Consequence

Provides a convenient global parameterization of convex bodies for reconstruction, computation of Minkowski sums, duality relations, and variational formulas; it translates geometric operations into convex analysis operations on functions.

Reversal

The radial (or gauge) function gives the distance from the origin to the boundary in direction u and is different in behaviour and domain of applicability; taking polars swaps support and Minkowski functional roles, reversing primal/dual perspectives.

Boundary

Defined for any nonempty subset via sup⟨x,u⟩, but classical support-function properties (convexity, positive homogeneity, uniqueness) presuppose K is closed and convex; for unbounded K values may be infinite for some directions.

Semantic Tension

Often conflated with the Minkowski functional (gauge) or with radial functions; while all are directional scalar descriptors of sets, they differ in homogeneity, duality relations, and sensitivity to translation.

Synthesis

The support function is the convex, positively homogeneous map that records the maximal directional projection of a convex body; it is a primary functional representation translating geometric constructs (sums, translations, duality) into tractable operations in convex analysis.