Support function

Definition

The support function h A : R n → R of a non-empty closed convex set A in R n is given by

x ∈ 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 is the (signed) distance of H(x) from the origin.

Examples

The support function of a singleton A={a} is h A ( x ) = x ⋅ a .

The support function of the Euclidean unit ball B₁ is h B 1 ( x ) = | x | .

If A is a line segment through the origin with endpoints -a and a then h A ( x ) = | x ⋅ a | .

As a function of x

The support function of a compact convex set is real valued and continuous, but if the set is unbounded, its support function is extended real valued (it takes the value ∞ ). 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 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

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).

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

and subadditive:

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 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 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:

and

The latter generalises to

where A + B denotes the Minkowski sum:

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

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 τ :A ↦ 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, τ 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 τ 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

In other words, for any x ∈ M , this support function gives the signed distance of the unique hyperplane that touches M in x.