In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the support function h_A of a non-empty closed convex set A in
describes the (signed) distances of supporting hyperplanes

Supporting hyperplane

Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set S in Euclidean space \mathbb R^n if it meets both of the following:...

 of A from the origin. The support function is a convex function on .
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.

Definition

The support function
of a non-empty closed convex set A in is given by
; 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 .

The support function of the Euclidean unit ball B₁ is .

If A is a line segment through the origin with endpoints -a and a then .

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 sinlgeton {y}, say, it follows that the support function is differentable at
u and its gradient coincides with y. Conversely, if h_A is differentiable at u, then A ∩ H(u) is a sinlgeton. Hence h_A is differentable 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

Convex function

In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...

.
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. 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 , 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

Hausdorff distance

In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right...

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