# Support function

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{{Short description|Distance from origin of tangent hyperplanes}}
{{distinguish|Support curve}}

In [mathematics](/source/mathematics), the '''support function''' ''h''<sub>''A''</sub> of a non-empty [closed](/source/Closed_set) [convex set](/source/convex_set) ''A'' in <math>\mathbb{R}^n</math>
describes the (signed) distances of [supporting hyperplane](/source/supporting_hyperplane)s of ''A'' from the origin. The support function is a [convex function](/source/convex_function) on <math>\mathbb{R}^n</math>.
Any non-empty closed convex set ''A''  is uniquely determined by   ''h''<sub>''A''</sub>. 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](/source/Minkowski_addition). 
Due to these properties, the support function is one of the most central basic concepts in [convex geometry](/source/convex_geometry).

==Definition==
The support function  <math>h_A\colon\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](/source/unit_vector): 
by definition, ''A'' is contained in the closed half space 
:<math>  \{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> 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''<sub>''A''</sub>(''x'') 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 <math>B = \{ y\in \mathbb{R}^n\,:\, \|y\|_2 \le 1\} </math> is  <math>h_{B}(x)=\|x\|_2</math> where <math>\|\cdot\|_2</math> is the 2-norm.

If ''A'' is a [line segment](/source/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'' nonempty convex set is real valued and continuous, but if the 
set is closed and unbounded, its support function is [extended real](/source/Extended_real_number_line) 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''<sub>''A''</sub> 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 ''h''<sub>''A''</sub>
is an [even function](/source/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''<sub>''A''</sub>'(''u'';''x'') denotes the [directional derivative](/source/directional_derivative) of
''h''<sub>''A''</sub> at ''u'' &ne; ''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'' &cap; ''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 ''h''<sub>''A''</sub> is differentiable at ''u'', then ''A'' &cap; ''H''(''u'') is a singleton. Hence ''h''<sub>''A''</sub> is differentiable at all points ''u'' &ne; ''0'' 
if and only if ''A'' is ''strictly convex'' (the boundary of ''A'' does not contain any line segments).

More generally, when <math>A</math> is convex and closed then for any <math>u\in \mathbb{R}^n\setminus\{0\}</math>,

:<math>\partial h_A(u) = H(u)\cap A\,, </math>

where <math>\partial h_A(u) </math> denotes the set of [subgradient](/source/subgradient)s of  <math>h_A</math> at <math>u</math>.

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''<sub>''A''</sub> is a [convex function](/source/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](/source/Legendre_transformation) of a positive homogeneous, convex, real valued function 
is the (convex) [indicator function](/source/Indicator_function_(convex_analysis)) of a compact convex set.

Many authors restrict  the support function to the Euclidean unit sphere 
and consider it as a function on ''S''<sup>''n''-1</sup>. 
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](/source/Minkowski_sum):
:<math>A + B := \{\, a + b \in \mathbb{R}^{n} \mid a \in A,\ b \in B \,\}.</math>
The [Hausdorff distance](/source/Hausdorff_distance) {{nowrap|''d''<sub>&thinsp;H</sub>(''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](/source/uniform_norm) on the [unit sphere](/source/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'' <sub>''A''</sub> 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](/source/convex_cone) of nonempty compact convex sets. 
The mapping <math>\tau</math> is an [isometry](/source/isometry) between this cone, endowed with the Hausdorff metric, and 
a subcone of the family of continuous functions on ''S''<sup>''n''-1</sup> with the uniform norm.

==Variants==
In contrast to the above, support functions are sometimes defined on the boundary of ''A'' rather than on 
''S''<sup>''n''-1</sup>, 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](/source/smooth_surface), ''M'', with a [unit normal vector](/source/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](/source/Barrier_cone)
* [Supporting functional](/source/Supporting_functional)

==References==
{{reflist}}

Category:Convex geometry
Category:Types of functions

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Adapted from the Wikipedia article [Support function](https://en.wikipedia.org/wiki/Support_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Support_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.