In mathematics, the support function hA of a non-empty closed convex set A in
describes the (signed) distances of supporting hyperplanes of A from the origin.
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.
[3] 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 word exterior is important here, as the orientation of x plays a role, the set H(x) is in general different from H(−x).
The support function of the Euclidean unit ball
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
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.
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).
It is crucial in convex geometry that these properties characterize support functions: Any positive homogeneous, convex, real valued function on
is the support function of a nonempty compact convex set.
The homogeneity property shows that this restriction determines the support function on
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.
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.
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.
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.
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.
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
, this support function gives the signed distance of the unique hyperplane that touches M in x.