130.203.167.237: /* Derivation */

2013-08-11T15:03:23Z

Derivation

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In [[mathematics]], the '''support function''' ''h''''A'' of a non-empty [[Closed set|closed]] [[convex set]] ''A'' in <math>\mathbb{R}^n</math>
describes the (signed) distances of [[supporting hyperplane]]s of ''A'' from the origin. The support function is a [[convex function]] on <math>\mathbb{R}^n</math>.
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 <math>h_A:\mathbb{R}^n\to\mathbb{R}</math>
of a non-empty closed convex set ''A'' in <math>\mathbb{R}^n</math> is given by
:<math> h_A(x)=\sup\{ x\cdot a: a\in A\},</math>
<math>x\in\mathbb{R}^n</math>; see
<ref name=bonnesen>T. Bonnesen, W. Fenchel, '' Theorie der konvexen Körper,'' Julius Springer, Berlin, 1934.
English translation: ''Theory of convex bodies,'' BCS Associates, Moscow, ID, 1987.</ref>
<ref name=gardner>R. J. Gardner, ''Geometric tomography,'' Cambridge University Press, New York, 1995. Second edition: 2006.</ref>
.<ref name=schneider>R. Schneider, ''Convex bodies: the Brunn-Minkowski theory,'' Cambridge University Press, Cambridge, 1993.</ref> Its interpretation is most intuitive when ''x'' is a unit vector:
by definition, ''A'' is contained in the closed half space
:<math> \{y\in\mathbb{R}^n: y\cdot x \le h_A(x) \}</math>
and there is at least one point of ''A'' in the boundary
:<math> 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 ''h''''A'' is the (signed) distance of ''H''(''x'') from the origin.

==Examples==
The support function of a singleton ''A''={''a''} is <math>h_{A}(x)=x \cdot a</math>.

The support function of the Euclidean unit ball ''B''''1'' is <math>h_{B_1}(x)=|x|</math>.

If ''A'' is a line segment through the origin with endpoints -''a'' and ''a'' then <math>h_A(x)=|x\cdot a|</math>.

==Properties==

===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
<math>\infty</math>). 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
:<math> 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 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:
:<math> h_A(\alpha x)=\alpha h_A(x), \qquad \alpha \ge 0, x\in \mathbb{R}^n,</math>
and subadditive:
:<math> h_A(x+y)\le h_A(x)+ h_A(y), \qquad x,y\in \mathbb{R}^n.</math>
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 <math>\mathbb{R}^n</math> is the
support function of a nonempty compact convex set. Several proofs are known
,<ref name=schneider/>
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 <math>\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> h_{\alpha A}(x)=\alpha h_A(x), \qquad \alpha \ge 0, x\in \mathbb{R}^n</math>
and
:<math> h_{A+b}(x)=h_A(x)+x\cdot b, \qquad x,b\in \mathbb{R}^n.</math>
The latter generalises to
:<math> 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>A + B := \{\, a + b \in \mathbb{R}^{n} \mid a \in A,\ b \in B \,\}.</math>
The [[Hausdorff distance]] {{nowrap|''d'' H(''A'', ''B'')}}
of two nonempty compact convex sets ''A'' and ''B'' can be expressed in terms of support functions,
: <math> 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>\tau</math>:''A'' <math>\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>\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>\tau</math> 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
: <math>{x}\mapsto{x}\cdot N({x})</math>.
In other words, for any <math>{x}\in M</math>, this support function gives the
signed distance of the unique hyperplane that touches ''M'' in ''x''.

== See also ==
* [[Barrier cone]]
* [[Supporting functional]]

==References==

<references/>

[[Category:Convex geometry]]
[[Category:Types of functions]]

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130.203.167.237: /* Derivation */