SupportFunktion

SupportFunktion, commonly called the support function, is a basic concept in convex analysis and geometric representation. For a nonempty set K ⊂ R^n, the support function h_K maps a direction u ∈ R^n to the maximal value of the linear form u^T x over x in K: h_K(u) = sup{x^T u : x ∈ K}. If K is bounded, h_K is finite for all u; if K is unbounded, h_K(u) may be infinite for some u.

Key properties include that h_K is positively homogeneous and subadditive when K is convex: h_K(tu) = t h_K(u) for t ≥ 0, and h_K(u + v) ≤ h_K(u) + h_K(v). If K is convex, closed, and bounded, h_K is continuous on R^n. The support function encodes K in the sense that K can be recovered as K = {x ∈ R^n : x^T u ≤ h_K(u) for all u ∈ R^n}. The polar set K° = {u ∈ R^n : h_K(u) ≤ 1} also relates to h_K, illustrating a duality between a set and its support function.

Common examples illustrate its geometry. If K is a ball of radius r centered at the origin, h_K(u) = r ||u||. If K is a convex polygon with vertices v_i, h_K(u) = max_i u^T v_i. For a singleton K = {c}, h_K(u) = u^T c.

Applications span optimization, duality, and computational geometry. The concept underpins representation of convex sets by support lines, dual norms, and Minkowski functionals.