# Support function

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

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.