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

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