In mathematics, the support function h_A of a non-empty closed convex set A in ${\displaystyle \mathbb {R} ^{n}}$ describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function on ${\displaystyle \mathbb {R} ^{n}}$. Any non-empty closed convex set A is uniquely determined by h_A. 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.

The support function ${\displaystyle h_{A}\colon \mathbb {R} ^{n}\to \mathbb {R} }$ of a non-empty closed convex set A in ${\displaystyle \mathbb {R} ^{n}}$ is given by

${\displaystyle x\in \mathbb {R} ^{n}}$; see . Its interpretation is most intuitive when x is a unit vector: by definition, A is contained in the closed half space

and there is at least one point of A in the boundary

of this half space. The hyperplane H(x) is therefore called a supporting hyperplane with exterior (or outer) unit normal vector 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). Now h_A(x) is the (signed) distance of H(x) from the origin.

The support function of a singleton A = {a} is ${\displaystyle h_{A}(x)=x\cdot a}$.

The support function of the Euclidean unit ball ${\displaystyle B=\{y\in \mathbb {R} ^{n}\,:\,\|y\|_{2}\leq 1\}}$ is ${\displaystyle h_{B}(x)=\|x\|_{2}}$ where ${\displaystyle \|\cdot \|_{2}}$ is the 2-norm.

If A is a line segment through the origin with endpoints −a and a, then ${\displaystyle h_{A}(x)=|x\cdot a|}$.