##  [Support Function](/support-function-0) 

 Definition

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

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

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

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

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

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

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

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

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.