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''support functional, of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s0912701.png" /> in a real vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s0912702.png" />''
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The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s0912703.png" /> on the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s0912704.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s0912705.png" />, defined by the relation
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s0912706.png" /></td> </tr></table>
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''support functional, of a set  $  A $
 +
in a real vector space  $  X $''
 +
 
 +
The function  $  sA $
 +
on the vector space  $  Y $
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dual to  $  X $,
 +
defined by the relation
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 +
$$
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( sA)( y)  = \sup _ {y \in A }  \langle  x, y\rangle.
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$$
  
 
For example, the support function of the unit sphere in a normed space considered in duality with its conjugate space is the norm in the latter.
 
For example, the support function of the unit sphere in a normed space considered in duality with its conjugate space is the norm in the latter.
  
A support function is always convex, closed and positively homogeneous (of the first order). The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s0912707.png" /> is a one-to-one mapping from the family of closed convex sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s0912708.png" /> onto the family of closed convex homogeneous functions; the inverse operator is the [[Subdifferential|subdifferential]] (at zero) of the support function. Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s0912709.png" /> is a closed convex subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s09127010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s09127011.png" />; and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s09127012.png" /> is a closed convex homogeneous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s09127013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s09127014.png" />. These two relations (resulting from the Fenchel–Moreau theorem, see [[Conjugate function|Conjugate function]]) also express the duality between closed convex sets and closed convex homogeneous functions.
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A support function is always convex, closed and positively homogeneous (of the first order). The operator $  s: A \rightarrow sA $
 +
is a one-to-one mapping from the family of closed convex sets in $  X $
 +
onto the family of closed convex homogeneous functions; the inverse operator is the [[Subdifferential|subdifferential]] (at zero) of the support function. Indeed, if $  A $
 +
is a closed convex subset in $  X $,  
 +
then $  \partial  ( sA) = A $;  
 +
and if $  p $
 +
is a closed convex homogeneous function on $  Y $,  
 +
then $  s( \partial  p( 0)) = p $.  
 +
These two relations (resulting from the Fenchel–Moreau theorem, see [[Conjugate function|Conjugate function]]) also express the duality between closed convex sets and closed convex homogeneous functions.
  
Examples of relations linking the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s09127015.png" /> with algebraic and set-theoretic operations are:
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Examples of relations linking the operator s $
 +
with algebraic and set-theoretic operations are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s09127016.png" /></td> </tr></table>
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$$
 +
s( \lambda C)  = \lambda sC, \lambda > 0; \ \
 +
s( A _ {1} + A _ {2} )  = sA _ {1} + sA _ {2} ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s09127017.png" /></td> </tr></table>
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$$
 +
s(  \mathop{\rm conv} ( A _ {1} \cup A _ {2} ))( x)  = \max ( sA _ {1} ( x), sA _ {2} ( x)).
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Minkowski,  "Geometrie der Zahlen" , Chelsea, reprint  (1953)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Minkowski,  "Gesammelte Abhandlungen" , '''2''' , Teubner  (1911)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Fenchel,  "On conjugate convex functions"  ''Canad. J. Math.'' , '''1'''  (1949)  pp. 73–77</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Fenchel,  "Convex cones, sets and functions" , Princeton Univ. Press  (1953)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L. Hörmander,  "Sur la fonction d'appui des ensembles convexes dans un espace localement convexe"  ''Ark. Mat.'' , '''3'''  (1955)  pp. 181–186</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.T. Rockafellar,  "Convex analysis" , Princeton Univ. Press  (1970)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Minkowski,  "Geometrie der Zahlen" , Chelsea, reprint  (1953)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Minkowski,  "Gesammelte Abhandlungen" , '''2''' , Teubner  (1911)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Fenchel,  "On conjugate convex functions"  ''Canad. J. Math.'' , '''1'''  (1949)  pp. 73–77</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  W. Fenchel,  "Convex cones, sets and functions" , Princeton Univ. Press  (1953)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L. Hörmander,  "Sur la fonction d'appui des ensembles convexes dans un espace localement convexe"  ''Ark. Mat.'' , '''3'''  (1955)  pp. 181–186</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Support functions play an important role in functional analysis and in other applications of convexity, e.g. optimization and geometry of numbers.
 
Support functions play an important role in functional analysis and in other applications of convexity, e.g. optimization and geometry of numbers.
  
Support functions of closed convex domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091270/s09127018.png" /> find application in the study of growth (and zero distribution) of entire functions, cf. e.g. [[Borel transform|Borel transform]]; [[Entire function|Entire function]]; [[Growth indicatrix|Growth indicatrix]].
+
Support functions of closed convex domains in $  \mathbf R  ^ {2n} $
 +
find application in the study of growth (and zero distribution) of entire functions, cf. e.g. [[Borel transform|Borel transform]]; [[Entire function|Entire function]]; [[Growth indicatrix|Growth indicatrix]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber,  C.G. Lekkerkerker,  "Geometry of numbers" , North-Holland  (1987)  pp. Sect. (iv)  (Updated reprint)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Schneider,  "Boundary structure and curvature of convex bodies"  J. Tölke (ed.)  J.M. Wills (ed.) , ''Contributions to geometry'' , Birkhäuser  (1979)  pp. 13–59</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber,  C.G. Lekkerkerker,  "Geometry of numbers" , North-Holland  (1987)  pp. Sect. (iv)  (Updated reprint)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Schneider,  "Boundary structure and curvature of convex bodies"  J. Tölke (ed.)  J.M. Wills (ed.) , ''Contributions to geometry'' , Birkhäuser  (1979)  pp. 13–59</TD></TR></table>

Latest revision as of 08:24, 6 June 2020


support functional, of a set $ A $ in a real vector space $ X $

The function $ sA $ on the vector space $ Y $ dual to $ X $, defined by the relation

$$ ( sA)( y) = \sup _ {y \in A } \langle x, y\rangle. $$

For example, the support function of the unit sphere in a normed space considered in duality with its conjugate space is the norm in the latter.

A support function is always convex, closed and positively homogeneous (of the first order). The operator $ s: A \rightarrow sA $ is a one-to-one mapping from the family of closed convex sets in $ X $ onto the family of closed convex homogeneous functions; the inverse operator is the subdifferential (at zero) of the support function. Indeed, if $ A $ is a closed convex subset in $ X $, then $ \partial ( sA) = A $; and if $ p $ is a closed convex homogeneous function on $ Y $, then $ s( \partial p( 0)) = p $. These two relations (resulting from the Fenchel–Moreau theorem, see Conjugate function) also express the duality between closed convex sets and closed convex homogeneous functions.

Examples of relations linking the operator $ s $ with algebraic and set-theoretic operations are:

$$ s( \lambda C) = \lambda sC, \lambda > 0; \ \ s( A _ {1} + A _ {2} ) = sA _ {1} + sA _ {2} ; $$

$$ s( \mathop{\rm conv} ( A _ {1} \cup A _ {2} ))( x) = \max ( sA _ {1} ( x), sA _ {2} ( x)). $$

References

[1] R.T. Rockafellar, "Convex analysis" , Princeton Univ. Press (1970)
[2] H. Minkowski, "Geometrie der Zahlen" , Chelsea, reprint (1953)
[3] H. Minkowski, "Gesammelte Abhandlungen" , 2 , Teubner (1911)
[4] W. Fenchel, "On conjugate convex functions" Canad. J. Math. , 1 (1949) pp. 73–77
[5] W. Fenchel, "Convex cones, sets and functions" , Princeton Univ. Press (1953)
[6] L. Hörmander, "Sur la fonction d'appui des ensembles convexes dans un espace localement convexe" Ark. Mat. , 3 (1955) pp. 181–186

Comments

Support functions play an important role in functional analysis and in other applications of convexity, e.g. optimization and geometry of numbers.

Support functions of closed convex domains in $ \mathbf R ^ {2n} $ find application in the study of growth (and zero distribution) of entire functions, cf. e.g. Borel transform; Entire function; Growth indicatrix.

References

[a1] P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a2] R. Schneider, "Boundary structure and curvature of convex bodies" J. Tölke (ed.) J.M. Wills (ed.) , Contributions to geometry , Birkhäuser (1979) pp. 13–59
How to Cite This Entry:
Support function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Support_function&oldid=48913
This article was adapted from an original article by V.M. Tikhomirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article