Support function
In mathematics, the support function hA of a non-empty closed convex set A in [math]\displaystyle{ \mathbb{R}^n }[/math] describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function on [math]\displaystyle{ \mathbb{R}^n }[/math]. 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
The support function [math]\displaystyle{ h_A\colon\mathbb{R}^n\to\mathbb{R} }[/math] of a non-empty closed convex set A in [math]\displaystyle{ \mathbb{R}^n }[/math] is given by
- [math]\displaystyle{ h_A(x)=\sup\{ x\cdot a: a\in A\}, }[/math]
[math]\displaystyle{ x\in\mathbb{R}^n }[/math]; see [1] [2] .[3] Its interpretation is most intuitive when x is a unit vector: by definition, A is contained in the closed half space
- [math]\displaystyle{ \{y\in\mathbb{R}^n: y\cdot x \leqslant h_A(x) \} }[/math]
and there is at least one point of A in the boundary
- [math]\displaystyle{ H(x)= \{y\in\mathbb{R}^n: y\cdot x = h_A(x) \} }[/math]
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 hA is the (signed) distance of H(x) from the origin.
Examples
The support function of a singleton A={a} is [math]\displaystyle{ h_{A}(x)=x \cdot a }[/math].
The support function of the Euclidean unit ball [math]\displaystyle{ B = \{ y\in \mathbb{R}^n\,:\, \|y\|_2 \le 1\} }[/math] is [math]\displaystyle{ h_{B}(x)=\|x\|_2 }[/math] where [math]\displaystyle{ \|\cdot\|_2 }[/math] is the 2-norm.
If A is a line segment through the origin with endpoints -a and a then [math]\displaystyle{ h_A(x)=|x\cdot a| }[/math].
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 [math]\displaystyle{ \infty }[/math]). 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
- [math]\displaystyle{ h_A'(u;x)= h_{A \cap H(u)}(x) \qquad x \in \mathbb{R}^n. }[/math]
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).
More generally, when [math]\displaystyle{ A }[/math] is convex and closed then for any [math]\displaystyle{ u\in \mathbb{R}^n\setminus\{0\} }[/math],
- [math]\displaystyle{ \partial h_A(u) = H(u)\cap A\,, }[/math]
where [math]\displaystyle{ \partial h_A(u) }[/math] denotes the set of subgradients of [math]\displaystyle{ h_A }[/math] at [math]\displaystyle{ u }[/math].
It follows directly from its definition that the support function is positive homogeneous:
- [math]\displaystyle{ h_A(\alpha x)=\alpha h_A(x), \qquad \alpha \ge 0, x\in \mathbb{R}^n, }[/math]
and subadditive:
- [math]\displaystyle{ h_A(x+y)\le h_A(x)+ h_A(y), \qquad x,y\in \mathbb{R}^n. }[/math]
It follows that hA is a convex function. It is crucial in convex geometry that these properties characterize support functions: Any positive homogeneous, convex, real valued function on [math]\displaystyle{ \mathbb{R}^n }[/math] is the support function of a nonempty compact convex set. Several proofs are known, [3] 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. The homogeneity property shows that this restriction determines the support function on [math]\displaystyle{ \mathbb{R}^n }[/math], 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:
- [math]\displaystyle{ h_{\alpha A}(x)=\alpha h_A(x), \qquad \alpha \ge 0, x\in \mathbb{R}^n }[/math]
and
- [math]\displaystyle{ h_{A+b}(x)=h_A(x)+x\cdot b, \qquad x,b\in \mathbb{R}^n. }[/math]
The latter generalises to
- [math]\displaystyle{ h_{A+B}(x)=h_A(x)+h_B(x), \qquad x\in \mathbb{R}^n, }[/math]
where A + B denotes the Minkowski sum:
- [math]\displaystyle{ A + B := \{\, a + b \in \mathbb{R}^{n} \mid a \in A,\ b \in B \,\}. }[/math]
The Hausdorff distance d H(A, B) of two nonempty compact convex sets A and B can be expressed in terms of support functions,
- [math]\displaystyle{ d_{\mathrm H}(A,B) = \| h_A-h_B\|_\infty }[/math]
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 [math]\displaystyle{ \tau }[/math]:A [math]\displaystyle{ \mapsto }[/math] 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, [math]\displaystyle{ \tau }[/math] 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 [math]\displaystyle{ \tau }[/math] 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
- [math]\displaystyle{ {x}\mapsto{x}\cdot N({x}) }[/math].
In other words, for any [math]\displaystyle{ {x}\in M }[/math], this support function gives the signed distance of the unique hyperplane that touches M in x.
See also
References
- ↑ T. Bonnesen, W. Fenchel, Theorie der konvexen Körper, Julius Springer, Berlin, 1934. English translation: Theory of convex bodies, BCS Associates, Moscow, ID, 1987.
- ↑ R. J. Gardner, Geometric tomography, Cambridge University Press, New York, 1995. Second edition: 2006.
- ↑ 3.0 3.1 R. Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.
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