Support function

Examples

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

The support function of the Euclidean unit ball ${\displaystyle B=\{y\in {\mathbb{R}}^n:\Vert y{\Vert }_2\le 1\}}$ is ${\displaystyle h_B(x)=\Vert x{\Vert }_2}$ where ${\displaystyle \Vert \cdot {\Vert }_2}$ is the 2-norm.

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

Properties

As a function of x

The support function of a compact nonempty convex set is real valued and continuous, but if the set is closed and unbounded, its support function is extended real valued (it takes the value ${\displaystyle \mathrm{\infty }}$). As any nonempty closed convex set is the intersection of its supporting half spaces, the function h_A determines A uniquely. This can be used to describe certain geometric properties of convex sets analytically. For instance, a set A is point symmetric with respect to the origin if and only if h_A is an even function.

In general, the support function is not differentiable. However, directional derivatives exist and yield support functions of support sets. If A is compact and convex, and h_A'(u;x) denotes the directional derivative of h_A at u ≠ 0 in direction x, we have

${\displaystyle h_A^{\prime }(u;x)=h_{A\cap H(u)}(x)x\in {\mathbb{R}}^n.}$

Here H(u) is the supporting hyperplane of A with exterior normal vector u, defined above. If A ∩ H(u) is a singleton {y}, say, it follows that the support function is differentiable at u and its gradient coincides with y. Conversely, if h_A is differentiable at u, then A ∩ H(u) is a singleton. Hence h_A is differentiable at all points u ≠ 0 if and only if A is strictly convex (the boundary of A does not contain any line segments).

More generally, when ${\displaystyle A}$ is convex and closed then for any ${\displaystyle u\in {\mathbb{R}}^n\setminus \{0\}}$,

${\displaystyle \mathrm{\partial }{h}_A(u)=H(u)\cap A,}$

where ${\displaystyle \mathrm{\partial }{h}_A(u)}$ denotes the set of subgradients of ${\displaystyle h_A}$ at ${\displaystyle u}$.

It follows directly from its definition that the support function is positive homogeneous:

${\displaystyle h_A(\alpha x)=\alpha h_A(x),\alpha \ge 0,x\in {\mathbb{R}}^n,}$

and subadditive:

${\displaystyle h_A(x+y)\le h_A(x)+h_A(y),x,y\in {\mathbb{R}}^n.}$

It follows that h_A is a convex function. It is crucial in convex geometry that these properties characterize support functions: Any positive homogeneous, convex, real valued function on ${\displaystyle {\mathbb{R}}^n}$ is the support function of a nonempty compact convex set. Several proofs are known, one is using the fact that the Legendre transform of a positive homogeneous, convex, real valued function is the (convex) indicator function of a compact convex set.

Many authors restrict the support function to the Euclidean unit sphere and consider it as a function on S^n-1. The homogeneity property shows that this restriction determines the support function on ${\displaystyle {\mathbb{R}}^n}$, as defined above.

As a function of A

The support functions of a dilated or translated set are closely related to the original set A:

${\displaystyle h_{\alpha A}(x)=\alpha h_A(x),\alpha \ge 0,x\in {\mathbb{R}}^n}$

and

${\displaystyle h_{A+b}(x)=h_A(x)+x\cdot b,x,b\in {\mathbb{R}}^n.}$

The latter generalises to

${\displaystyle h_{A+B}(x)=h_A(x)+h_B(x),x\in {\mathbb{R}}^n,}$

where A + B denotes the Minkowski sum:

${\displaystyle A+B:=\{a+b\in {\mathbb{R}}^n\mid a\in A,b\in B\}.}$

The Hausdorff distance d _H(A, B) of two nonempty compact convex sets A and B can be expressed in terms of support functions,

${\displaystyle d_{\mathrm{H}}(A,B)=\Vert h_A-h_B{\Vert }_{\mathrm{\infty }}}$

where, on the right hand side, the uniform norm on the unit sphere is used.

The properties of the support function as a function of the set A are sometimes summarized in saying that ${\displaystyle \tau }$:A ${\displaystyle \mapsto }$ h _A maps the family of non-empty compact convex sets to the cone of all real-valued continuous functions on the sphere whose positive homogeneous extension is convex. Abusing terminology slightly, ${\displaystyle \tau }$ is sometimes called linear, as it respects Minkowski addition, although it is not defined on a linear space, but rather on an (abstract) convex cone of nonempty compact convex sets. The mapping ${\displaystyle \tau }$ is an isometry between this cone, endowed with the Hausdorff metric, and a subcone of the family of continuous functions on S^n-1 with the uniform norm.

Variants

In contrast to the above, support functions are sometimes defined on the boundary of A rather than on S^n-1, under the assumption that there exists a unique exterior unit normal at each boundary point. Convexity is not needed for the definition. For an oriented regular surface, M, with a unit normal vector, N, defined everywhere on its surface, the support function is then defined by

${\displaystyle x\mapsto x\cdot N(x)}$.

In other words, for any ${\displaystyle x\in M}$, this support function gives the signed distance of the unique hyperplane that touches M in x.