Support function

support functional, of a set in a real vector space

The function on the vector space dual to , defined by the relation

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 is a one-to-one mapping from the family of closed convex sets in onto the family of closed convex homogeneous functions; the inverse operator is the subdifferential (at zero) of the support function. Indeed, if is a closed convex subset in , then ; and if is a closed convex homogeneous function on , then . 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 with algebraic and set-theoretic operations are:

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 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