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''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|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  $  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====
<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====
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|Borel transform]]; [[Entire function|Entire function]]; [[Growth indicatrix|Growth indicatrix]].

====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>