In mathematics, the support function hA of a non-empty closed convex set A describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function. 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

of a non-empty closed convex set A is given by

Interpretation

Unit vectors

is a unit vector is contained in the closed half-space

in the boundary

is therefore a supporting hyperplane. 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 represents the signed physical Euclidean distance.

Non-unit vectors as scoring rates

no longer represents a direct spatial distance. Instead, it can be interpreted as a total score governed by a scoring.

from the origin to the supporting hyperplane is determined by dividing the maximum accumulated score by the scoring rate:

purely as an artifact of the adjusted measuring criteria.

Examples

The support function of a singleton A = {a.

is the 2-norm.

If A is a line segment through the origin with endpoints −a and 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). As any nonempty closed convex set is the intersection of its supporting half spaces, the function hA 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 hA 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 hA'(u;x) denotes the directional derivative of hA 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 hA is differentiable at u, then A H(u) is a singleton. Hence hA 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).

,

denotes the set of subgradients.

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

and subadditive:

It follows that hA is a convex function 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 Sn-1, 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 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:A h A 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 is an isometry between this cone, endowed with the Hausdorff metric, and a subcone of the family of continuous functions on Sn-1 with the uniform norm.

Variants

In contrast to the above, support functions are sometimes defined on the boundary of A rather than on Sn-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

1. .

, this support function gives the signed distance of the unique hyperplane that touches M in x.

See also

• Barrier cone

• Supporting functional

References