In mathematics, the support function hA of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}}
describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function on R n {\displaystyle \mathbb {R} ^{n}} . Any non-empty closed convex set A is uniquely determined by hA. Furthermore, the support function, as a function of the set A, is compatible with many natural geometric operations, like scaling, translation, rotation and Minkowski addition. Due to these properties, the support function is one of the most central basic concepts in convex geometry.
Definition The support function h A : R n → R {\displaystyle h_{A}\colon \mathbb {R} ^{n}\to \mathbb {R} } of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}} is given by
h A ( x ) = sup { x ⋅ a : a ∈ A } , {\displaystyle h_{A}(x)=\sup\{x\cdot a:a\in A\},}
Interpretation
Unit vectors The interpretation of the support function is most intuitive when x {\displaystyle x} is a unit vector. By definition, the convex set A {\displaystyle A} is contained in the closed half-space
{ y ∈ R n : y ⋅ x ⩽ h A ( x ) } {\displaystyle \{y\in \mathbb {R} ^{n}:y\cdot x\leqslant h_{A}(x)\}}
and there is at least one point of A {\displaystyle A} in the boundary
H ( x ) = { y ∈ R n : y ⋅ x = h A ( x ) } {\displaystyle H(x)=\{y\in \mathbb {R} ^{n}:y\cdot x=h_{A}(x)\}}
of this half-space. The hyperplane H ( x ) {\displaystyle H(x)} is therefore a supporting hyperplane with exterior unit normal vector x {\displaystyle x} . The word exterior is important here, as the orientation of x plays a role, the set H(x) is in general different from H(−x). In this specific case, h A ( x ) {\displaystyle h_{A}(x)} represents the signed physical Euclidean distance from the origin to the supporting hyperplane H ( x ) {\displaystyle H(x)} .
Non-unit vectors as scoring rates For an arbitrary, non-unit vector x ≠ 0 {\displaystyle x\neq 0} , the support function value h A ( x ) {\displaystyle h_{A}(x)} no longer represents a direct spatial distance. Instead, it can be interpreted as a total score achievable on A {\displaystyle A} governed by a scoring determined by x {\displaystyle x} . In this framework, the vector x {\displaystyle x} functions as an evaluation operator where each element acts as a weighting factor. The inner product x ⋅ a {\displaystyle x\cdot a} is a linear machine that converts a spatial position vector a ∈ A {\displaystyle a\in A} into a scalar score. The magnitude | | x | | 2 {\displaystyle ||x||_{2}} defines the sensitivity of this machine, representing how many score units are accumulated per meter of physical displacement in the direction of x {\displaystyle x} . Under this interpretation, the support function value h A ( x ) {\displaystyle h_{A}(x)} is the maximum possible score that can be achieved by any point within the set A {\displaystyle A} along the direction x {\displaystyle x} . The supporting hyperplane H ( x ) = { y ∈ R n : x ⋅ y = h A ( x ) } {\displaystyle H(x)=\{y\in \mathbb {R} ^{n}:x\cdot y=h_{A}(x)\}} represents the collection of all points in space that achieve this exact maximum score threshold. The true physical distance d {\displaystyle d} from the origin to the supporting hyperplane is determined by dividing the maximum accumulated score by the scoring rate:
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